Skip to main content

Full text of "An introduction to the theory of relativity"

See other formats


Engineering 
- 



AN INTRODUCTION TO 
THE THEORY OF RELATIVITY 



V RELATIVITY: THE SPECIAL AND THE GENERAL 
THEORY. By ALBERT EINSTEIN, Ph.D. Translated by 
R. W. LAWSON. Fifth edition. Crown 8vo. 5s. net. 

OTHER BOOKS ON THE EINSTEIN THEORY. 
V SPACE-TIME-MATTER. By HERMANN WEYL. Demy 8vo. 

21s. net. 

EINSTEIN THE SEARCHER: His Work Explained in 
Dialogues with Einstein. By ALEXANDER MOSZKOWSKI. 
Demy 8vo. 12s. 6d. net. 

AN INTRODUCTION TO THE THEORY OF RELA- 
TIVITY. By LYNDON BOLTON, M.A. Crown 8vo. 

5s. net. 

V RELATIVITY AND GRAVITATION. By Various Writers. 
Edited by J. MALCOLM BIRD. Crown 8vo. 7s. 6d. net. 

RELATIVITY AND THE UNIVERSE. By Dr. HARRY 
SCHMIDT. Crown 8vo. 7s. 6d. net. 

V THE IDEAS OF EINSTEIN'S THEORY. (The Theory of 
Relativity in Simple Language.) By J. H. THIRRING, 
Ph.D. Crown 8vo. 5s. net. 

THE FOURTH DIMENSION SIMPLY EXPLAINED. 
A Collection of Essays Selected from those Submitted in 
The Scientific American Prize Contest. Edited by HENRY 
P. MANNING. Crown 8vo. 7s. 6d. net. 



AN INTRODUCTION 

TO THE 

THEORY OF RELATIVITY 



BY 

L. BOLTON, M.A. 

FORMERLY A SCHOLAR OF CLARE COLLEGE, CAMBRIDGE 
A SENIOR EXAMINER IN H.M. PATENT OFFICE 



dyeameVojjTOS (uriTW. PLATO (alleged) 



WITH 38 DIAGRAMS 



METHUEN & GO. LTD. 

36 ESSEX STREET W.C. 

LONDON 






&*; 

: 



First Published t in igai 



PREFACE 

r I A HE Theory of Relativity may very well prove 
-* to be the most important single contribution 
yet made to intellectual thought. If the theory 
is true it means nothing less than that physical 
science has at length broken through the crust of 
the phenomenal and apparent. The mechanism of 
nature is to be sought in something as yet con- 
ceivable only mathematically. 

It is not to be expected that a theory of this 
novelty and scope can be other than difficult. No 
one can be surprised if he finds the general drift 
hard to grasp. This was indeed by far the most 
serious difficulty encountered by the writer. It is 
nothing but literal fact that he found it a greater 
obstacle to a general understanding of the subject 
than the details of the advanced mathematical 
work. Try as he would the drift eluded him. 
The main, almost the sole, object of the present 
book is to meet this difficulty, all other considera- 
tions being subordinate. He has written the book 
with a very lively recollection of his own troubles, 

V 

515413 



vi THE THEORY OF THE RELATIVITY 

and he hopes that it may be of service to others in 
like case. 

The great question is, What is it all about ? To 
this question some give one answer and some 
another ; but none, to the writer's knowledge, give 
so clear an answer as Einstein himself,* and even he 
answers it by implication rather than directly. Still 
the implication of his definitions of the Special, or 
Restricted, and of the General principles is so plain 
that there is no mistaking it. His definition of the 
Restricted Principle, which need not be given here, 
as it is fully dealt with in the following pages, is a 
compendium of the special theory and it is easily 
generalized. His definition of the General Principle 
simply repeats the definition of the Restricted Prin- 
ciple in wider terms, and he makes it quite clear 
thatCRelativity as a whole is the theory of the state- 
ment of general physical laws informs common to all 
observers^ It is something of a puzzle why other 
writers of authority have not given this fact a more 
prominent place and stated it plainly and explicitly. 
It may have been because it seemed so obvious as 
not to require emphasis, but to the writer's mind 
the greater part of the mystery which has sur- 
rounded the subject has arisen through failure to 
grasp it. It was certainly so in his own case. 

* " Relativity, the Special and the General Theory." By Albert 
Einstein, Ph.D. Translated by R. W. Lawson. Fifth Edition. 
Crown 8vo, 53. net. (Methuen & Co., Ltd.) 



PREFACE vii 

When he realized it, the whole subject, till then a 
hopeless jig-saw puzzle, seemed to arrange itself of 
its own accord. The " Scientific American," in 
their remarks on the award of the Eugene Higgins 
Prize,* which the writer was fortunate enough to 
win, were good enough to compliment him on the 
" extraordinarily fine judgment which he used in 
deciding just what he would say and what he would 
leave unsaid ". As a matter of fact, what he did 
was to say what was strictly relevant to this main 
issue and side-track what was not. He hopes that 
whatever the shortcomings of the present book 
may be, he has at least left the reader's mind clear 
on this all-important point. 

The writer, therefore, has followed Einstein in 
this general conception of the subject ; though the 
treatment differs very considerably in detail. ^The 
book is, to a large extent, the winning essay extended 
to twelve or thirteen times its length.) The object is 
to show that the conclusions of the subject develop 
easily and naturally out of the search for a general 
mode of statement of physical laws. All matter 
which is not strictly relevant to this end is either 
omitted altogether, or where the amount of public 
attention which has been directed to certain points 
forbids their exclusion, it is expressly stated that 

*An account of this contest together with a selection of the 
essays and other matter is being issued under the title " Relativity 
and Gravitation ". Crown 8vo, 73. 6d. net. (Methuen & Co. , 
Ltd.) 



viii THE THEORY OF THE RELATIVITY 

the discussion is a digression (see Chapter X). No 
attempt at exhaustive, or indeed wide treatment, 
has been made. Necessarily, in order to say 
what Relativity is about, considerable detail is re- 
quired, but nothing has been introduced beyond 
what is absolutely necessary to this end. 

The present book departs from the essay in one 
important respect. Mathematical symbols are used 
with considerable freedom. The writer contrived 
to avoid them in the essay ; but, while writing it, 
he was conscious all the time that he was thinking 
mathematics, and that his exposition, such as it 
was, suffered by the absence of symbols. It is 
impossible to avoid mathematics, and the motto on 
the title-page is meant to imply this fact. It is the 
notice which Plato is alleged to have put up warn- 
ing indifferent mathematicians off his premises. 
Physical laws must be stated in mathematical terms 
to be of any value, and the subject is therefore 
essentially mathematical. To expect a non-mathe- 
matical treatment of Relativity is as reasonable as 
to expect a non-mathematical treatment of the In- 
tegral Calculus. At the same time, a very small 
amount of mathematical knowledge indeed is re- 
quired for a general grasp of the subject. The 
mathematical knowledge assumed in this book is 
exiguously small. Einstein says that his book 
presumes a standard of education corresponding 
to that of a university matriculation examination. 



PREFACE ix 

The present book, the writer thinks, requires less, 
nothing in fact beyond simple equations and Euclid 
I, 47 (the Theorem of Pythagoras). Wherever a 
proof is given it is written out in great detail, and 
this may at first sight give the impression of over- 
much mathematics. This extreme detail may be 
unnecessary, but the writer felt that it was better 
to be on the safe side. 

Perhaps the most serious difficulty after that of 
understanding the drift of the subject is the neces- 
sity for getting rid of all metaphysical notions. 
Philosophic questions may be considered at the 
end of the subject, but at the beginning and in 
the course of the subject they are out of place 
and misleading. The difficulty of suppressing 
metaphysical considerations is of a peculiarly in- 
sidious kind, and it requires a distinct mental effort 
to overcome it. Particular attention should there- 
fore be paid to what is said in the text on this 
point. 

Next to this is the necessity for understanding 
the nature of reference frames and systems, and 
their relation to an observer's point of view. It is 
hoped that Chapter 1 1 1 will clear up this important 
matter. 

There is one matter of detail in which the prize 
essay has been departed from, and that is the treat- 
ment of rotation (chapter XIII). In the essay 
the writer borrowed his illustration respecting 

b 



x THE THEORY OF THE RELATIVITY 

measured times and lengths on a rotating system 
from Einstein, and he still thinks that Einstein's 
illustration is most apt and telling when properly 
understood. Unfortunately, experience has proved 
to him that it raises so many irrelevant suggestions 
as to make it practically useless. He has therefore 
most reluctantly abandoned it. 

Naturally, a very large number of books and 
other publications have been laid under contribu- 
tion, and the writer gratefully admits his obligation 
to the authors. He has endeavoured to do full 
justice in the way of acknowledgment ; but if 
there are any omissions he hopes the authors will 
realize how impossible it is to acknowledge every 
detail. 

In conclusion, the writer desires to thank the 
friends who have helped him by their criticisms 
and suggestions. Their help has been invaluable 
to him. He also desires to thank Mr. F. E. Smith 
of Bedford School for drawing the diagrams. 

L. B. 

BEDFORD, 23 May, 1921. 



CONTENTS 



PAGE 



Preface v 

I. Introductory - ..... i 

II. Metaphysics, Physics, and Mathematics - - 10 

III. Reference Systems - - 17 

IV. Velocity, Acceleration, Mass, and Momentum- 31 
V. Physical Laws - - 34 

VI. The Mechanical Principle of Relativity - - 39 

VII. The Lorentz Transformation - 47 

VIII. The Velocity of Light 51 

IX. The Restricted Principle of Relativity - - 58 

X. Some Special Features of the Restricted Theory 

of Relativity - 70 

XI. The Four-dimensional Continuum - - 84 

XII. The General Principle of Relativity - 96 

XIII. Rotating Systems - - -^ 101 

XIV. Translation - ----109 
XV. Natural Gravitational Fields - - - -113 

XVI. Geometry of the Gravitation Theory - - 119 

XVII. Geometry (continued) - - 131 

XVIII. The Gravitation Theory - - - - 136 

XIX. The Crucial Phenomena - - - - 148 

XX. The Application of the General Principle - 156 

XXI. General Summary and Conclusion - - 162 

Bibliographical Note - 173 

Index - - 175 
xi 



AN INTRODUCTION TO THE 
THEORY OF RELATIVITY 

CHAPTER I 
INTRODUCTORY 

>^ 

IF the reader expects a non-mathematical treatment 
of the Theory of Relativity, it is to be feared that he 
will be disappointed. The subject matter belongs to 
mathematical physics ; indeed, in a sense, it includes 
the whole of mathematical physics, for it deals with the 
mathematical expression of those descriptive state- 
ments of fact which are called physical laws. It is, 
therefore, impossible to avoid mathematics ; we must 
think more or less in mathematical terms, even though 
no symbols are actually written down. Fortunately, 
the amount required for a general understanding of 
the subject is very small, and should present no diffi- 
culty, with the full explanations which it is proposed 
to give. The real obstacle in the way of a generally 
intelligible treatment is not so much the fact that 
mathematics is necessarily involved as the unfamiliar 
character of the subject matter, which renders it some- 
what difficult to give at the start a clear indication of 
its drift. It is not difficult to devise a form of words 
which substantially covers the ground, but the words 



2 THE THEORY OF RELATIVITY 

themselves gather their meaning from the subject, and 
are interpreted differently according to the reader's 
knowledge of it. The observations immediately follow- 
ing, therefore, must not be taken as exhaustive, but it 
is thought that they will give a reasonably clear pre- 
liminary idea of the kind of subject matter with which 
we shall have to deal. 

^Phenomena look different to different people, though 
the phenomenon or thing is physically the same) It is 
unthinkable that an observer can change the nature 
of anything by merely looking at it. Now, Relativity 
seeks to reconcile these differences and to determine 
statements of fact which shall be independent of 
different observers ; which shall describe phenomena 
independently of any particular point of view. \Rela- 
tivity is the theory of the expression of general physical 
facts in a way which shall be common to all observers 
and independent of anyone in particular,} Looked at 
in this way, " Relativity " is not altogether a satis- 
factory name. It concentrates attention too much on 
individual points of view, whereas the real object is 
their elimination. The point is not without import- 
ance. The Relativity of Knowledge is a well-known 
philosophical doctrine, and the name Relativity mis- 
leads some persons, more especially if they have an 
acquaintance with metaphysics, into the belief that 
Einstein's theory is nothing more than a reassertion 
of the doctrine in a slightly modified shape.* It is 

* Since this paragraph was written, Lord Haldane's book, " The 
Reign of Relativity," has appeared. This work deals compre- 
hensively with the Relativity of Knowledge. Amongst other 
things the position of Einstein's theory more especially those 



INTRODUCTORY 3 

probably too late now to choose another name, but it 
is as well to remember that the name Relativity savours 
of the lucus a non lucendo principle. 

It is not the writer's purpose to anticipate subsequent 
discussions by enlarging upon previous attempts at the 
statements of general physical laws, upon the limita- 
tions of these attempts, or upon the suppositions which 
underlie them. These will appear in due course, but 
meanwhile as an example it may be stated that until 
the relativists interfered it had always been thought 
that in specifying objects or phenomena, measurements 
of space were to be treated as entirely distinct and 
independent from those of time, and that general 
physical laws, that is to say, general statements of 
fact independent of particular observers, could be 
framed on that basis. The following instance illus- 
trates the relativist position in regard to this important 
matter, but the points raised will be dealt with in 
greater detail later on. 

Let us take some simple physical object such as a 
cube, which we shall suppose to be opaque and to have 
its edges less than the distance between the pupils of 
the observer's eyes. The qualifications are of no great 
importance, but they simplify the discussion somewhat. 
If now we take our station so that our eyes are as nearly 
as possible opposite the middle of one of the faces, what 
we see is simply a square as in the first diagram of 
Fig. i. As we move round to the left, a second face 
comes into view, the top and bottom sides of the two 
faces losing their apparent parallelism, and tending to 

parts of it relating to space and time measurements in relation to 
this doctrine is discussed. 



4 THE THEORY OF RELATIVITY 

vanishing points according to the rules of perspective. 
We do not get quite the same impression from both 
eyes, since they are at different station points, and this, 
together with differences in illumination, the presence 
of intervening objects, and, above all, previous experi- 
ence, gives us a sense of relief or solidity depending 
upon our distance away from the cube. As we move 
still further round and occupy other points of view, we 
get the impressions shown by the other diagrams of 




FIG. i. 

the figure, until when we come opposite the next face 
of the cube, we see a square as at first ; and so on. 
We get yet another set of impressions by first taking 
our stand in a position corresponding to diagram No. 3 
of Fig. i, and then moving upwards so that our eyes, 
when in the next position, are level with the top of 
the cube. The succession of impressions as we move 
round over the top is shown in Fig. 2, ending up with 
a square set diamond-wise, which we see when we are 



INTRODUCTORY 5 

looking down on the cube from some point over the 
middle of the top. We never see the whole cube at 
once ; we can, in fact, never see more than three faces 
at the same time, but by combining the impressions 
got by occupying a number of station points, we are 
able to form an opinion as to its shape. As the physicist 
would say, we construct a theory of its shape. We 
say that it is a solid figure, square in plan, front eleva7 




FIG. 2. 

tion and side elevation. We could have made the cube 
go through another series of aspects by approaching it 
or moving further away ; and, of course, we could have 
made it present itself in all its aspects merely by turning 
it round, and moving it nearer or further off. These 
modifications are, however, immaterial. The point is 
that we get an idea of what the cube is as a physical 
object by looking at it in various aspects from different 
view points and collating the results. 



6 THE THEORY OF RELATIVITY 

Now, in obtaining these aspects we have moved right 
or left, up or down, and backwards or forwards. All 
our view points have been arrived at by moving in one 
or more of these three mutually perpendicular direc- 
tions. We have, in fact, constructed our theory of 
physical shape on the supposition that we are dealing 
with a three-dimensional object in three-dimensional 
space, and no other considerations have entered our 
minds. 

It is hardly necessary to point out that though an 
engineer would proceed by measurement, and thus 
construct his theory in a more refined way, his method 
would amount to the same as ours. Indeed, nothing 
else is to be expected, seeing that even the vaguest 
impressions of magnitude are in fact measurements, 
rough indeed, but still comparisons with things we have 
seen before. 

At this point the relativist steps in. He says that 
the set of aspects presented by viewing the cube from 
different stations in space are insufficient to give a 
correct theory of its physical shape. He says that yet 
another set would be presented if the cube and the 
observer were in relative motion. If the observer were 
to keep on the move while collecting his impressions, 
or making his measurements, if this were possible, these 
would be different from the impressions or measure- 
ments he would get while standing still. For example, 
if during the first observation, when the observer saw 
the cube as a square, he had been moving through his 
point of observation sideways, parallel with the face, 
he would have observed the figure, not as a square, 
but as a parallelogram (Fig. 3), having its width less 



INTRODUCTORY 



than its height. In fact, all the different figures would 
have been crushed up, like an accordion, in the direction 
of motion. It is true, says the relativist, that this 
crushing is imperceptible, but this, so he says, is because 
the observer's motion is comparatively slow. If it 
were of the order of the velocity of light, the change of 
shape would be manifest, and if it were usual for things 
to move about at such a rate, these changes of shape 
would be accepted as matters of course. In fact, 
electrons, which sometimes move with very high 
velocities, though their magnitudes are too small for 



FIG. 3. 

direct observation of their changes of shape, do actually 
exhibit peculiarities of motion which can be shown to 
be the direct consequences of these changes. 

The relativist goes on to say that his statements, so 
far from causing surprise or incredulity, are only what 
we ought to expect. By introducing velocity into our 
theory of shape, we have only done what nature always 
does, and brought in the element of time. Time and 
space are never separated in nature, and we have no 
right to separate them in our theories which are sup- 
posed to represent nature. Things exist both in space 



8 THE THEORY OF RELATIVITY 

and time, and the two are inseparably joined.* Any 
physical theory of the shape of a thing must therefore 
include time. It is wrong to consider a thing as existing 
in three-dimensional space on the one hand, and endur- 
ing independently in one-dimensional time on the other. 
He sometimes puzzles us by saying that an object must 
be regarded as a four-dimensional thing existing in a 
four-dimensional continuum, as he calls this combina- 
tion of time and space, but all he means is that we must 
take into account in a certain special manner a fourth 
element, namely, time, in addition to length, breadth, 
and height, or thickness. If we ask the relativist 
whether he thinks that space and time are the same, 
he says, " What I said was that they are inseparable, 
not that you should not distinguish them. If you like 
to go into the subject further you will find that your 
mathematical processes will distinguish them for you 
quite sufficiently for your purposes." He tells us that 
he, no more than we, can picture four-dimensional 
objects, and, furthermore, that he does not want to do 
it, and that it would not help him much for physical 
purposes if he could, seeing that he can get all he 
requires merely by supposing things to be determined 
by four independent quantities instead of three. But 
of this more hereafter. In subsequent chapters we 
shall see what evidence the relativist can produce in 
support of his strange theories. 

Meanwhile there are some preliminary matters 
requiring attention. A number of words have been 
used whose meaning is probably not at all clear to the 

* " Einstein's Theories of Relativity and Gravitation " (Scientific 
American Publishing Co., New York ; also Methuen & Co., Ltd., 
London), p. 186. 



INTRODUCTORY 9 

reader. We have spoken of physics, metaphysics, 
physical laws, mathematics, space, time, points of view, 
and so forth ; all of which require definition. It is 
very unlikely, for example, that the reader knows what 
is meant by a point of view, which is probably the most 
important for our purposes of any. In the next four 
chapters we shall deal with these matters and some 
others required for the subsequent work. 



Summary. Relativity treats of the mathematical 
expression of general physical laws. For a general 
understanding of the subject, mathematical ideas are 
required, but no great proficiency. The difficulty of 
the theory resides in its novelty. Space and time are 
inseparable ; for example, no physical theory of shape 
can be framed which excludes time. 



CHAPTER II 

METAPHYSICS, PHYSICS, AND MATHEMATICS 
i. METAPHYSICS 

THE " New English Dictionary " defines meta- 
physics as " that branch of speculative inquiry 
which treats of the first principles of things, including 
such concepts as being, substance, essence, time, space, 
cause, identity, etc. " ; " theoretical philosophy as the 
ultimate science of Being and Knowing ". The name 
seems to have referred originally merely to the order 
in which the books dealing with these subjects occurred 
in the received edition of Aristotle's writings. These 
books came after those on physics. By a misinterpre- 
tation the preposition ^e-ra acquired the meaning of 
beyond or transcending, which now attaches to it in 
this connexion. 

Now, seeing that these inquiries go to the very root 
and essence of things, it would seem only proper and 
logical to take their results as the foundation of all 
other knowledge. Unfortunately, though they have 
engaged the attention of many of the greatest minds 
from the earliest ages, and are still pursued, no definite 
conclusions have been reached. The inner nature of 
space, time, cause, and such like concepts still remains 
undefined, and those who wish to pursue other branches 

10 



METAPHYSICS, PHYSICS, AND MATHEMATICS n 

of knowledge must therefore find some starting point 
other than metaphysics. 

2. PHYSICS 

The physicist accordingly defines things as they 
present themselves to his observation. We are mainly 
concerned here with the fundamental concepts, space 
and time, so we will take these as examples. The 
physicist does not know what space is ; the meta- 
physician has not told him. But he defines a length, 
or the distance between two points, on a plane for 
instance, to be the number of times a given standard 
or unit measuring rod will have to be laid down end- 
ways in successive adjacent positions along the straight 
line joining the two points so as to reach from one to 
the other. If the points are not on a plane, but on a 
curved surface, such as that of the earth, he cannot 
proceed far in a straight line, and if necessary he 
modifies this definition in an obvious way. If his 
measuring rod is what he calls a yard, and he has to 
lay down the rod so many times, he says the distance 
between the points, or the length of the interval 
between them, is the same number of yards. He 
assumes that his standard measure is rigid, that is, that 
it does not alter its own length capriciously without 
his knowledge. If it did so alter, his measurements 
would be at fault unless everything else altered accord- 
ingly, in which case he would have no optical means 
of knowing it, though mechanical means might per- 
haps be available. So also for time. An interval of 
time between two events is for his purposes the number 
of rotations or the fraction of a rotation of the earth, 



t2 THE THEORY OF RELATIVITY 

the number of oscillations of a pendulum, or the number 
of vibrations of a sodium atom, or the like, which take 
place between the occurrence of the two events. In 
measuring time he assumes a quality corresponding to 
rigidity in his standard lengths ; he assumes that his 
clocks, as we may call his time-pieces, whatever their 
nature, do not alter their rates without his knowing it. 

Time and space in their physical sense are thus 
intervals of time and measured lengths. Physical time 
and space are entirely distinct from the concepts of 
duration and extension of the metaphysician. They 
are time and space as disclosed by measurement, or, if 
the expression be preferred, they are the behaviour of 
clocks and measuring rods. 

This distinction between physical and metaphysical 
time and space is all-important in the present subject. 
If it is not clearly understood that these words are used 
solely in their physical sense unless otherwise stated, 
most of what follows will sound paradoxical, or even 
nonsensical. Much of the misunderstanding of the 
theory of relativity would be avoided if this distinction 
were kept in mind. 

This seemingly arbitrary way of denning things 
without previously investigating their nature may be 
thought to be unsound and liable to error. There are 
two answers to this. In the first place, whatever may 
be the whole content of the concept of time or of space, 
measured time or measured length is part of it, so that 
the physical sense cannot be wrong ; at the worst it 
can only be inadequate. In the next place, whether 
actually wrong or merely inadequate, experiment gives 
means for checking it. Sooner or later deduction based 



METAPHYSICS, PHYSICS, AND MATHEMATICS 13 

on the definitions will lead to results which experiment 
will contradict, and thus show the need for amend- 
ment. This procedure was not open to the early 
philosophers, who did not understand the method of 
experiment. More will be said on this point in a later 
chapter when we come to deal with the method of 
hypothesis. It is sufficient for the present to observe 
that so far the physical definitions have not led to any 
such contradictions. 

It is, of course, not necessary to assume that the 
physical definitions comprise the whole content of the 
concepts of space and time. Some hold that they do, 
but this opinion is repugnant to many minds. For 
physical purposes the point is immaterial as long as 
the results are consistent with experiment. 

In addition to definitions, the physicist requires 
certain fundamental principles in the form of postu- 
lates or axioms. These can be treated on similar lines 
to the definitions, and checked by experiment. 

3. MATHEMATICS. 

The mathematician meets the metaphysical difficulty 
in a different way. He simply says that it is not his 
business to inquire whether his definitions and postu- 
lates are accurate representations of things or not. As 
long as they are not self -contradictory and are mutually 
consistent they satisfy his requirements. Physical 
truth and mathematical truth are different things. 
The definitions and postulates of physics have to agree 
with nature, those of mathematics need only agree 
with one another. The truth of Euclid would be 



14 THE THEORY OF RELATIVITY 

unaffected though such things as squares, straight lines, 
right angles, and the like never existed. Indeed, it is 
very unlikely that they do exist. The chances are 
probably millions to one against the existence of an 
exact square according to Euclid's definition, and it is 
quite certain that no one is gifted with faculties refined 
enough to recognize it if it did exist. As a rule, mathe- 
matical definitions agree with natural conditions more 
or less, since they are generally suggested by them, 
but it is not necessary that they should, and the mathe- 
matician, if he is a pure mathematician and not a 
physicist, is not concerned. 

4. MATHEMATICAL PHYSICS 

It follows that when mathematical processes are 
applied to physics, the provisional assumption is made 
that the definitions and postulates of the mathe- 
matician are applicable to physical phenomena. If 
this assumption is incorrect the mathematical deduc- 
tions disagree with experimental tests. There have 
been cases in which disagreement has arisen and has 
led to important discoveries. 

One of the best instances of this is Planck's quantum 
theory of radiation. Matter is perpetually radiating 
energy in the form of pulses or waves. These waves 
vary in length, from the almost infinitesimal propor- 
tions of the X-rays and possibly still smaller rays as 
yet undetected up to those of wireless rays, the lengths 
of which may extend to miles and no one knows how 
much bigger. In between are the rays which manifest 
themselves as light and heat. If these rays strike other 



METAPHYSICS, PHYSICS, AND MATHEMATICS 15 

bodies they are absorbed, reflected or transmitted. 
All these rays represent so much energy, and they 
might be made to do mechanical work by proper 
appliances. If now we consider any closed region from 
which no radiation can escape and which none can 
enter, a continual exchange in the form of radiation 
takes place between the various bodies present, absorbed 
rays being in turn emitted in the form of other rays, 
and thus a state of equilibrium is eventually reached in 
which all the radiations exactly balance, and no further 
apparent physical change occurs It had been assumed 
that this emission of energy took place in accordance 
with electro-magnetic laws which involve complete 
continuity when applied to the mechanism of radiation 
and absorption that is to say, that the mathematical 
definition of continuity was satisfied within the limits 
of experimental error, so that the mathematical pro- 
cesses founded upon this definition could be applied to 
all the circumstances of the case. It was found, how- 
ever, that this supposition involved the concentration 
of an infinite amount of energy in the aether, which 
would thus drain all the energy out of the bodies in the 
enclosure, and this was contrary to experience. It 
was, therefore, assumed that energy was emitted by 
small jumps which are integral multiples of minute 
definite quanta, each quantum being proportional 
to the wave-length of the emitted radiation. This 
supposition agrees with the facts. It is not found 
necessary to suppose that the absorption takes place 
otherwise than continuously. 

It is very improbable that the agreement between 
mathematical definitions and axioms, and physical 



1 6 THE THEORY OF RELATIVITY 

facts, is ever complete, but if the discrepancy is within 
the limits of experimental error it is undetectable. 



Summary. Metaphysics deals with the fundamental 
nature of things, but it has not led to results sufficiently 
definite for the purposes of physics and mathematics. 
The physicist takes things as he observes them, trusting 
to experiment to correct him. The mathematician is 
not concerned with physical existence, but only with 
consistency. The application of mathematics to 
physics involves hypothesis. The relativist concepts 
of space and time are physical, not metaphysical. 



CHAPTER III 
REFERENCE SYSTEMS 

THIS chapter deals with the implications of the 
term " point of view ". 

If we wish to form an accurate idea with the aid of 
a map of a tract of country at which we are look- 
ing, a good way is to lay on the map a piece of 
thin transparent celluloid ruled with intersecting lines. 







FIG. 4. 

The lines, as shown in Fig. 4, should preferably intersect 
at right angles and divide the celluloid strip into 
squares, and they should be as numerous as possible, 
consistent with clearness. We select a pair of these 
lines, such as the thick ones in the figure, and apply 
the strip to the map so that their intersection registers 
with the spot on the map corresponding to our point of 
view, orienting the scale in any convenient way. If 
2 17 



i8 



THE THEORY OF RELATIVITY 



now we imagine the landscape covered with a network 
of lines corresponding to those on the scale, and simi- 
larly situated, we have a reference frame whereby we 
may determine the positions of any of the objects in 
view relative to one another and to ourselves. 

The same principle might be applied in other ways. 
For example, the strip might be replaced by a circular 
disk, Fig. 5, marked with equidistant concentric circles 
and lines radiating from their common centre O. We 
might apply this disk to the map as before, and imagine 
the landscape to be divided up similarly. 




FIG. 5. 

It is evident that this procedure can be varied inde- 
finitely according to convenience. Even the equality 
of the divisions is not absolutely essential ; it makes 
measurements easier, but that is all. If measurements 
are not required, but only relative positions, irregular 
lines will often answer our purpose as well as others. 
The essentials are that some sort of reference frame 
corresponding to every point of view taken up is 
required, and that once a reference frame has been 
selected it must be supposed to remain rigid. If it 



REFERENCE SYSTEMS 



alters its configuration it is of no use. A reference 
frame is the natural correlative of an observers point 
of view. The one implies the other. 

The physicist also has to determine and record 
positions. He observes or determines occurrences in 
what he calls fields of force that is to say, regions in 
which various agencies gravitational, electric, mag- 
netic, mechanical, cohesive, and so on are acting, 
together with the effects of these agencies. He pro- 
ceeds in precisely the same way as the observer in the 
case of the landscape, and fits out his field with a rigid 



M 



FIG. 6. 



reference frame which represents his point of view. 
The physicist calls his reference frames " co-ordinates " 
or " systems of co-ordinates ". 

We proceed to give particulars of the various systems 
of co-ordinates which will be used in this book. 

i. CARTESIAN CO-ORDINATES 

The name of this system is derived from the philo- 
sopher Descartes, who invented it. The principle is 
the same as that described in connexion with Fig. 4. 
We shall first describe the system with reference to a 
plane, and afterwards extend it to three dimensions. 



20 THE THEORY OF RELATIVITY 

Referring to Fig. 6, Ox and Oy are two straight lines 
at right angles, intersecting at a point 0, which is 
called the origin. All other points, such as P, are 
located with reference to these two lines, called the 
axes. Ox is called the axis of x, and Oy is the axis oiy. 
It does not matter to which of the lines we give these 
names, but they are usually applied as shown. The 
position of any point, P, is determined when we know 
the lengths of the perpendiculars, PM, PN dropped 
from P upon them, or the lengths of the lines ON, and 
OM, which are equal to PM and PN each to each. PM 
or ON is called the ordinate of P, and PN or OM is called 
the abscissa of P, and the ordinate and abscissa of a 
point are collectively called its co-ordinates. Thus, if 
PN is three units long and PM two units, we say that 
the co-ordinates of P are 3 and 2, or that P is the 
point (3, 2), always writing the abscissa first. If we 
do not wish to particularize the point numerically we 
use letters and call it the point (a, b), or if its position 
is subject to continual change, as, for instance, when 
we are considering a point moving along some straight 
line such as AB, we usually go to the end of the alphabet 
and call it the point (x, y). Thus we should say, " Let 
(x, y) be any point P on the straight line AB ". 

A system of axes such as has just been described, 
intersecting at right angles, is called " rectangular 
axes," but it is sometimes convenient to use " oblique 
axes " where Ox and Oy do not intersect at right angles, 
as in Fig. 7. The abscissa PA 7 or OM, and the ordinate 
PM or ON, are always taken parallel to the axes. The 
system of nomenclature is the same as before. 

It is easy to adapt the Cartesian system to three 



REFERENCE SYSTEMS 



21 



dimensions by taking a third axis Oz through the 
origin, perpendicular to both Ox and Oy, as in Fig. 8, 
which is a perspective view. The three pairs of axes, Oy 




FIG. 7. 

and Oz, Oz and Ox, Ox and Oy, now define three planes, 
like three sides of a box which meet at a corner 0, 
and the axes Ox, Oy, and Oz are the edges of the box, 




meeting at the same corner. The position of any 
point, P, in space, is determined by its perpendicular 
distances, PL, PM, and PN from these planes. Con- 



22 THE THEORY OF RELATIVITY 

formably with the notation used already we may call 
the point P (a, b, c) or (x, y, z), as the case may be, 
using any convenient letters. The three distances, PL, 
PM, and PA 7 are each called ordinates (the word abscissa 
is not used in solid geometry) ; collectively they are 
called co-ordinates, as before. If we complete the box- 
like figure having the origin at one end of a diagonal 
and the point P at the other, we can see that there 
are four lines equal to each of the co-ordinates of P, 
like the edges of a brick. 

Oblique axes are seldom or never used in solid 
geometry, and they need not detain us. 

There is one slight difference between the reference 
frame of the physicist and that which we supposed the 
observer of the landscape to use. The reference frame 
of the latter consisted not merely of two mutually 
perpendicular datum lines, but also of a number of 
others parallel to them at regular intervals. There is 
no real change in principle. The physicist could proceed 
in exactly the same way, but it is not always necessary. 
All that is usually needed is to put in any subsidiary 
lines by measurement as they are wanted, instead of 
supposing them to be drawn beforehand like the rulings 
on squared paper. The abscissae and ordinates of points 
are these subsidiary lines drawn ad hoc. A similar 
remark applies to three-dimensional reference frames. 
A reference frame for three dimensions, if completed, 
would be a mass of three sets of lines of indefinite length, 
the members of each set being parallel respectively to 
the three'axes, and perpendicular to those of the other 
two sets. 



REFERENCE SYSTEMS 23 

2. POLAR CO-ORDINATES 

These are the same as the second kind of reference 
frame considered in our imaginary survey of a tract of 
country. In this case we choose, as in Fig. 9, some 
convenient datum line, OA, which is called the initial 
line, and a point in it corresponding to the origin in 
Cartesian co-ordinates, which is called the pole. Any 
point P is determined by its distance OP from 0, and 
by the angle POA between the lines OP and OA. 
OP, which is usually denoted by the letter r, is called 
the radius vector, and the angle POA, usually written 0, 




FIG. 9. 

the vectorial angle. The polar co-ordinates of P are 
thus r and 6, or P is called the point (r, 6), always 
writing the radius vector first. When r is given, the 
point P obviously must lie on a circle PB, having r as 
radius and centre 0. This circle corresponds to one of 
the circles which we considered in connexion with 
Fig. 5, though it is never the practice to draw it. OP 
corresponds to one of the radial lines in Fig. 5. It is 
obvious that polar co-ordinates are nothing more than 
the range and bearing of the artilleryman, being the 
gun position, P the target, OP, or r, the range, and 6 the 
bearing from the zero line OA. 

In polar co-ordinates for three dimensions a second 



24 THE THEORY OF RELATIVITY 

angle is used. Let P, Fig. 10, be a point in space, 
the pole, OA the initial line. Take any fixed line 
OZ perpendicular to OA. To fix our ideas, suppose 
that OA is in a horizontal plane, and OZ vertical ; but 
the actual directions are immaterial, as long as the lines 
are mutually perpendicular. Drop a perpendicular, 
PM, from P on to the horizontal plane through 0, and 
join OM. Let be the angle which OP makes with 
the vertical OZ, and let $ be the angle which OM makes 
with OA. $ is therefore in the horizontal plane and 9 




FIG. 10. 

in a vertical plane. Let r be the length of the radius 
vector OP. Then the polar co-ordinates of P are 
(r, 6, (f>) written in that order. Polar co-ordinates thus 
correspond to range, angle of sight, and bearing in 
gunnery, the difference being that the angle of sight, 
a, is measured from the horizontal plane, and 6 from 
the vertical line, so that = 90 - a. If be the 
centre of the earth and P a point on its surface, < is 
the longitude of P, and a the latitude. It is clear that 
a system of parallels of latitude and meridians of longi- 
tude is a reference system adapted to a sphere. 



REFERENCE SYSTEMS 25 

3. GAUSSIAN CO-ORDINATES 

The straight lines and circles of which the foregoing 
reference frames are composed are only particular or 
limiting cases of curves. A perfectly general form of 
reference frame would therefore consist of sets of curves, 
as in Fig. n, which represents such a reference frame 
for two dimensions of space. The curves may be on 
any surface, the surface of the earth for example. 
These systems are called Gaussian co-ordinates, after 
Gauss, the mathematician, who first used them. 




Gaussian co-ordinates consist of a set of curves A 
indefinite in number, drawn according to any regular 
plan, and crossing a similar set B. All the curves A 
intersect the curves B, but none of the curves of either 
set intersect those of the same set. They must be 
capable of covering the surface on which they are 
drawn continuously that is to say, if P and P f are 
two points near one another, it must be possible to 
draw separate curves of one or both sets through the 
points, however close together they may be. A good 



26 THE THEORY OF RELATIVITY 

idea of this system of co-ordinates may be gathered 
by imagining it to be a network whose meshes differ 
but slightly in shape from neighbouring ones, but the 
difference may mount up considerably when the meshes 
are far apart. Any point P is located by the inter- 
section of two curves of different sets. Thus, in Fig. n, 
P is at the intersection of the curves 4 of set A and 3 of 
set B. Though the curves are drawn according to some 
regular plan, successive ones do not generally occur at 
equal intervals, and the system, therefore, does not 
lend itself to measurement as with the systems pre- 
viously considered. It is only useful for embodying 
the highly generalized ideas of mathematicians. We 
shall hear of it again later on. 

The extension to three-dimensional space is obvious. 
We have only to imagine a third set of curves inter- 
secting the members of the other two, but not on the 
same surface. The system may be pictured as three 
sets of threads imbedded in a solid. 

The physicist must assign times as well as places to 
his phenomena. This fact is of sufficient importance 
in the theory of relativity to require a special nomen- 
clature. The position of an occurrence is called a 
point. Its position and time taken together are called 
a point-event, or shortly, an event. 

To determine events every reference frame must be 
supposed to be filled with time indicating apparatus, 
called clocks for brevity, whatever their nature. Every 
point must theoretically have its clock. In practice 
we need suppose clocks to be only where they are 
wanted, just as lines in a reference frame were drawn 



REFERENCE SYSTEMS 27 

only where they were wanted. But wherever the clocks 
are, they must be set together and must go at the same 
rate, otherwise events could not be related definitely 
to one another. 

The synchronization of two clocks is not so simple a 
matter as it might appear at first sight. If the two 
clocks are at the same place it can be perceived directly 
whether they are together or not. But if they are in 
different places some signalling method has to be 
arranged, and the meaning of synchronism will depend 
upon this system. If the clocks are of the usual con- 
struction, an obvious method is for the observer to 
place himself midway between them and arrange 
mirrors so that he can see their faces at the same time. 
That is to say, a system of light signalling is used. 
Accordingly, whatever the construction of the clocks, 
their indications are supposed to be signalled by light 
to a point midway between them, and if these signals 
reach an observer at that point simultaneously the 
clocks are said to be in synchronism. A succession of 
such observations determines whether they are going 
at the same rate or not. This definition of syn- 
chronism in terms involving light is an important 
point in the theory of relativity. 

We shall call a reference frame with its clocks a 
reference system, and we shall use the words observer's 
system omitting the word reference to mean the 
observer himself, his reference system, his laboratory, 
and, in fact, all the objects which share his state of 
rest or motion. 



28 THE THEORY OF RELATIVITY 

A difficulty is sometimes felt, not less by those who 
are accustomed to theoretical work in which co-ordinate 
systems are used than by others, in realizing that 
reference frames are anything more than paper dia- 
grams. Paper diagrams are only representations of 
reference frames. In the sense in which the term is 
used in this book, a reference frame is to be taken as 
the systematized form of the ideas of distance and 
direction which every one is constantly using more or 
less vaguely and unconsciously. Thus the reference 
frame of a geographer consists of lines of latitude and 
longitude, the latitude of a place being measured by 
the angular distance of the place from the equator, and 
the longitude by the angle which a meridian circle 
through the north and south poles and the place makes 
with a similar circle drawn through some given place, 
such as Greenwich. The addition of sea-level gives 
him a complete three-dimensional reference frame. 
The reference frame of an astronomer is constructed 
on the same principle. There are three such in use. 
If, like the geographer, he takes the equator (or, rather, 
the trace of the plane of the equator produced to meet 
an imaginary celestial sphere) as his fundamental circle, 
he fixes any point by " declination " and " right 
ascension ". Declination corresponds to terrestrial 
latitude, and right ascension to longitude, excepting 
that the fixed point corresponding to Greenwich is a 
point on the equator called the " first point of Aries," 
which is the place where the sun crosses the equator 
from south to north at the Vernal Equinox. The 
meridians of right ascension are drawn through the 
north and south poles. Right ascension is always 



REFERENCE SYSTEMS 29 

reckoned towards the east, and usually in hours, 
minutes, and seconds of time. Astronomers also use 
celestial latitude and longitude, but these do not cor- 
respond to terrestrial latitude and longitude. Accord- 
ing to this system the fundamental circle is the ecliptic 
the path of the sun in the heavens and the meridians 
of longitude are drawn through the poles of this circle 
that is, points 90 away from it. Within recent years 
a system of co-ordinates called " galactic latitude and 
longitude " has come into use. In these the Milky Way 
is used as a fundamental circle. A physicist uses all 
sorts of frames, the walls of his laboratory, or the 
arrangement of his instruments. The simplest measure- 
ment requires some sort of reference frame, or system 
if we include clocks. If it is only a matter of reading 
a mercury barometer or a thermometer a reference 
frame consisting of one line merely the tube of the 
instrument is used. Regarded in this way, a refer- 
ence system is a real thing, and the origin may be 
taken as corresponding to what we call our point of 
view. 

The subject matter of relativity is the correlation of 
aspects of things obtained from different points of view, 
and its object is to investigate the conditions under 
which it is possible to describe things in ways which 
will apply to different reference systems that is to 
say, to inquire into the possibility of obtaining state- 
ments of fact which will hold good when one reference 
system is exchanged for another. 

General rules exist for changing over, or transforming, 
as it is called, from one reference system to another, 
but the examples of tranformatiori which will concern 



30 THE THEORY OF RELATIVITY 

us are very simple, and they can be dealt with as they 
arise. 



Summary. Reference frames are means for locating 
positions in space. The most usual forms are Cartesian 
and Polar co-ordinates. Cartesian co-ordinates consist 
of intersecting sets of parallel straight lines or planes. 
Polar co-ordinates are practically the same thing as 
range, angle of sight, and bearing. Gaussian co-ordi- 
nates include the others as particular cases, and consist 
of a network of curved lines arranged continuously : 
they do not as a rule lend themselves to measurement. 
A reference system is a reference frame plus clocks. 
The clocks are synchronized by light signals. Refer- 
ence systems are to be looked upon as really existing, 
and are not to be confounded with diagrams drawn on 
paper. A point of view is the origin of a reference 
system of some sort or other. 



CHAPTER IV 

VELOCITY, ACCELERATION, MASS, AND 
MOMENTUM 

IN the subsequent work reference will be made to 
velocity, acceleration, mass, and momentum. We 
proceed to give short explanations of these terms. 

i. VELOCITY 

If a body moves through a distance I in time t, its 
average velocity throughout the time t is lit. If the 
motion is uniform so that equal distances are described 
in equal times, this average velocity is the same as its 
actual or instantaneous velocity at any instant during 
the time t. If the motion is not uniform this is no 
longer the case, but the smaller the interval t the more 
nearly uniform is the motion and the more nearly does 
the average velocity throughout the interval approach 
the actual velocity at any instant during the interval. 
By taking time intervals short enough we can approxi- 
mate as closely as we please to instantaneous velocities. 

The term velocity implies more than mere speed ; 
it involves the direction of displacement. Velocity is 
speed in a given direction, which direction must be 
specified. In this book we shall use the term speed 
instead of velocity when it is not required to take 
direction into account. 



32 THE THEORY OF RELATIVITY 

2. ACCELERATION 

When the velocity of a body is changing either in 
magnitude or direction, the body is said to be acceler- 
ated, and any such change is always the result of the 
application of force. Without force there can be no 
acceleration. Every body possesses the property called 
inertia that is to say, it persists in a state of rest, or 
of uniform motion in a straight line, unless acted on 
by a force. 

The word acceleration is used in a technical sense in 
mechanics to cover not only increase, but decrease of 
speed with or without change of direction, and also 
change of direction without change of speed. If the 
force causing the acceleration is in the same straight 
line as the velocity of a moving body, the speed only 
varies but not the direction ; if it acts perpendicularly 
to the direction of the velocity, this direction is changed 
but no change of speed results. Intermediate direc- 
tions of application of a force produce both effects 
combined. 

If a body is rotating about an axis, every particle 
in it, excepting those on the axis, is subject to accelera- 
tion in the technical sense of the word, even though the 
rotation is uniform, since the direction of motion of 
each particle is continually changing though the speed 
may be constant. 

Frequent mention will be made in what follows of 
accelerated systems of reference. This term will there- 
fore be understood to include systems which are moving 
with varying speed without rotation, and also rotating 
systems in which the speed of the parts is not necessarily 
changing. 



VELOCITY, ACCELERATION, MASS, ETC. 33 

3. MASS 

Mass is one of the fundamental physical quantities, 
like space and time, of which no satisfactory meta- 
physical definition can be given. It is sometimes said 
to be quantity of matter. If a piece of material 
possesses a certain mass, then a piece of the same 
material of double the volume will have double the 
mass under the same physical conditions. The masses 
of different bodies are proportional to their weights, if 
they are weighed in the same locality. Mass, however, 
is not the same thing as weight, for if a body be weighed 
at sea level and on a mountain top, or at the equator 
and one of the poles, its weight will differ in each case 
on account of its altered distance from the centre of 
the earth, though the quantity of matter remains the 
same. 

4. MOMENTUM 

This is the name given to the product of the mass of 
a body into its velocity. Thus, if m be the mass of a 
body and v its velocity, the momentum is mv. As 
mass is sometimes said to be quantity of matter, so 
momentum is said to be quantity of motion, though the 
word motion is often used in the simple sense of dis- 
placement without any idea of mass attaching to it* 



CHAPTER V 
PHYSICAL LAWS 

THE general physical laws of nature are statements 
in compact form of uniformities which experience 
has shown to exist amongst physical phenomena. 
They have no compelling or binding power like the 
laws of the land or Divine dispensations. They are 
simply generalized statements of what has been found 
to happen in given circumstances, and may therefore 
be expected to happen in the future in like circum- 
stances. 

Prior to the development of experimental science, 
which is of comparatively modern origin, physical laws 
were derived from metaphysical considerations. Thus 
the heavenly bodies were assumed by the ancients to 
move in circles, on the ground that nature was perfect 
and the circle was a perfect figure ; no motion other 
than circular, therefore, was compatible with the per- 
fection of nature. The absence of any adequate experi- 
mental means of checking physical laws threw the 
whole burden of their proof on to the soundness of 
their premises. Hence the supreme importance of 
metaphysical inquiry which alone, through pure reason, 
could be looked to for the groundwork. 

The rise of the experimental method has altered all 

34 



PHYSICAL LAWS 35 

this by making it possible to apply the method of 
hypothesis to the establishment of these laws. Some 
circumstance, or set of circumstances, makes it probable 
that a certain generalization holds good. It is, there- 
fore, assumed provisionally as an hypothesis, and 
deductions are drawn from it. If these deductions, 
when tested by experiment, agree with observation, 
then the assumption reaches a higher degree of proba- 
bility, which may rise to practical certainty if it gives 
an explanation of phenomena previously unexplained, 
and still more, if it has to its credit the prediction of 
new phenomena. For example, Newton assumed as 
an hypothesis his law of gravitation. He tried it on 
the moon, but owing to an inaccurate estimate of the 
moon's distance his calculations did not agree with 
observation. Some time afterwards, with the aid of 
more exact figures, he got concordant results. This 
went a long way towards establishing the theory. 
Newton himself, and many mathematicians after him, 
notably Laplace, applied the theory to the other 
bodies of the solar system, and it was found to explain 
practically all their movements, and even to predict 
the existence of an important new planet. The theory 
was then regarded as proved. 

Its agreement with observation is astonishingly close. 
The only discrepancy of any importance is a small 
irregularity in the motion of the planet Mercury upon 
which it is not necessary to dwell at this stage, as it 
will be treated in greater detail later on. For the 
present it is enough to observe that Einstein's theory 
gives an adequate explanation, and it now seems clear 
that Newton's law of gravitation is only a first approxi- 
mation to the truth, though an exceedingly close one. 



36 THE THEORY OF RELATIVITY 

The method of hypothesis has sometimes been stigma- 
tized as mere guessing. This is unfair and foolish. 
An hypothesis is, of course, in the first instance very 
often a guess, but it is the sort of guess of which only 
talent and knowledge are capable, amounting at times 
to a flash of genius little short of inspiration. Those 
who use this language ignore the meticulous pains 
which are taken to verify by experiment the deductions 
from the hypotheses before they are accepted as laws. 
Einstein's theory is probably the finest instance on 
record of an inspired guess. 

Natural laws do not " explain " anything in the 
widest sense of the word. They tell us what happens, 
but not how or why it happens. If when referring to 
physical laws such words as " cause," " because," 
" therefore," and the like, are used, no philosophical or 
metaphysical theory of the efficiency of causation is 
implied. A cause in physics is merely an antecedent 
set of circumstances found by experience invariably to 
precede another set which is called the effect. If, for 
example, we say that the velocity of a body is increas- 
ing because a force is acting, no reason is implied why 
forces should so act. All that is meant is that in past 
instances it has been found that acceleration is always 
preceded by the application of force, and to suppose 
that anything different is occurring in the present case 
is inconsistent with experience. Explanation, in the 
physical sense, is merely grouping together separate 
happenings into one general statement.* 

* A very clear and full discussion of the scientific meaning of the 
word "explanation" is given in Herbert Spencer's "First Prin- 
ciples," Part I, Chapter IV. 



PHYSICAL LAWS 37 

The most important general feature of physical laws 
is that they are capable of mathematical expression ; 
in fact, they require it. This distinguishes physical 
laws from such general statements as the law of supply 
and demand and other economic laws, Grimm's law, 
and various others belonging to the less exact sciences. 
Physical laws are used for exact deductions and 
numerical computation, and mathematical expression 
is essential. In this book, whenever reference is made 
to a general physical law, its mathematical expression 
will be understood. 

As an example of the mathematical expression of a 
physical law, consider Newton's second law of motion, 
which is one of the postulates of mechanics. Newton 
himself calls it an " axiom ". The law states that the 
change of motion of a body is proportional to the force 
impressed upon it, and takes place in the direction of 
the force. By " change of motion " is meant change 
of momentum, which was defined in the last chapter, 
and the " force impressed " means the product of the 
force into the time during which it acts. If, therefore, 
m be the mass of the body, v the velocity in the direction 
of the force F at the time when it commences to act, 
and v' the velocity at the end of a time t t then the 
change in momentum is mv' mv, or m(v' v). 
This, the law says, is proportional to Ft, or equal to 
this product if units be properly chosen, so that 

m(v' - v) = Ft 
or, 

-r V' V 

F = m - 
Now (v 1 v)/t is the change in velocity per unit 



38 THE THEORY OF RELATIVITY 

time, which, in continuation of what was said on the 
subject in the last chapter, may be denned to be 
acceleration. Call this acceleration /, and we have 

F = mf 

as the mathematical expression of Newton's second 
law. As a particular case, we may take that of a 
heavy body whose weight is W, and mass m. W is thus 
the force with which the earth attracts the body. If 
g is the acceleration, that is to say, the velocity, 
produced in one second by the earth's attraction, which 
is 32 feet per second, we have W = mg. 

General physical laws are general in two ways. They 
must apply not only to large numbers of particular 
physical facts, but also to the circumstances of large 
numbers of particular observers. A law which is 
peculiar to the circumstances of one or a few observers 
only cannot be said to be general. Statements of 
physical laws must, therefore, as far as possible be 
independent of the points of view of particular ob- 
servers ; in other words, the forms of their mathe- 
matical expressions should be independent of any 
particular system of reference. This condition will be 
examined further in the next chapter. 



Summary. General physical laws are compact state- 
ments of uniformities. They are established by the 
method of hypothesis checked by experiment. They 
do not imply any metaphysical theory of causation. 
To be of any value they must be expressed mathematic- 
ally. They should permit of statement in identical 
form for different observers. 



CHAPTER VI 
THE MECHANICAL PRINCIPLE OF RELATIVITY 

PHYSICAL laws relate to measurable phenomena 
such as velocities, accelerations, forces, and the 
like located in particular places at particular times, 
and, therefore, in accordance with what we have seen 
in Chapter III, they require to be stated in relation to 
some reference system. In the last chapter we saw 
that for complete generality, they should be stated in 
such forms as are common to all observers. In the 
present chapter we shall inquire into the method of 
complying with this condition, confining the discussion 
to the laws of the Newtonian mechanics the classical 
mechanics, as the subject has been called. 

An observer's system must either be accelerated or 
unaccelerated, the latter term including a state of rest. 
We may at once rule out accelerated systems as unsuit- 
able for the statement of general mechanical laws, for 
unless all the systems were subject to the same accelera- 
tion a condition which is obviously impracticable 
the different accelerations and the corresponding forces 
which we have seen always accompany them, would 
have to be taken into account, each system having 
accelerations and forces peculiar to it. Now, forces 
and accelerations enter into the statements of the laws 

39 



40 THE THEORY OF RELATIVITY 

of Newtonian mechanics, as, for example, the second 
law of motion, which was considered in the last 
chapter. Consequently, if accelerated systems are used, 
the statements are complicated by accelerations and 
forces peculiar to each system. The possibility of 
framing any statement common to all is therefore 
precluded. 

We have therefore to fall back on unaccelerated 
systems, and the question resolves itself into a choice 
of co-ordinates. To examine generally the various 
reference frames which present themselves would lead 
us too deeply into mathematics, and we shall therefore 
content ourselves for the present with saying that 
mechanical laws are usually stated with reference to 
unaccelerated rectangular Cartesian co-ordinates, or 
Galilean co-ordinates as they are called after Galileo, 
the founder of the modern science of mechanics. It 
has been found that when so stated general mechanical 
laws preserve their mathematical form whatever may 
be the relative motion between observers. In other 
words, when stated with reference to the system of one 
observer, they may be stated in exactly similar mathe- 
matical form with reference to that of any other 
moving relatively to him provided only that this motion 
is unaccelerated that is to say, as the reader will 
remember, uniform in magnitude and direction and 
without rotation. We proceed to illustrate this, taking 
as an example Newton's second law of motion. For 
simplicity the investigation will be confined to one 
plane only, in which all the movements will be sup- 
posed to take place. 

Suppose an observer stationed at (Fig. 12), and 



MECHANICAL PRINCIPLE OF RELATIVITY 41 



using the Cartesian reference frame Ox, Oy, to observe 
a particle P of mass m. By " particle " we understand 
a body which has mass, but the magnitude of which is 
so small that it need not be taken into account. Let 
the particle P be supposed to be already in motion 
parallel to Ox at the commencement of the observation, 
and let its velocity increase by an amount V during 
the time of observation t, under the influence of a 
force F, also parallel to Ox. 

Let a second observer 0' ', using the Cartesian re- 
ference system 0V, O'y f such that 0V slides along 
Ox with uniform velocity u, observe the same particle 



FIG. 12. 

during the same time t, and let him apply the second 
law of motion to the particle. At the commencement 
of the time t we shall suppose that he observes its 
velocity to be v. Its momentum is therefore mv. At 
the end of the time its velocity, according to 0', is 
v -f V, since V ', being a gain in velocity, is assumed to 
be the same for both observers. The momentum is, 
therefore, m(v + V). Thus the change of momentum, 
according to the measurements of 0', is m(v -f V) - 
mv, or simply mV. This change the second law of 
motion declares to be equal to Ft, and thus 

F =- mVjt. 



42 THE THEORY OF RELATIVITY 

Now let go through a corresponding process. 0' is 
moving relatively to with a velocity u, and since at 
the commencement of the time t the particle is moving 
relatively to 0' with a velocity v, its velocity, according 
to O's measurements, is u -f v, and its momentum is 
m(u -f z;). Similarly, at the end of the time t, its 
momentum is m(u + v + V)> and its change of momen- 
tum is m(u + v -\- V) m(u + v) or raF. As in the 
case of 0', puts this equal to Ft, and thus obtains 

F = w7/*, 

which is exactly the same as the statement at which 0' 
arrived. It will thus be seen that the velocity u, which 
constitutes the difference between O's point of view 
and that of 0', has cancelled out, enabling them both 
to state the whole circumstances of the problem in 
identical form. 

This would not have been possible had the relative 
velocity u between and 0' not been uniform. Sup- 
pose that in the time / it had altered from u to u', and 
also that according to 0' the particle had gained a 
velocity V. Then the change of momentum, according 
to 0', would be mV. But according to it would be 

m(v + V + u '} m (v + ) 
or, 

m{V + (u f - u)} 

and O's statement of the second law would have been 

m ( u ' "~ u ) 



which is not identical with that of 0'. 
It is thus plain that the possibility of the existence 



MECHANICAL PRINCIPLE OF RELATIVITY 43 

of identical statements of the law hangs upon the fact 
that the relative velocity is uniform. This is an essen- 
tial condition. 

The principle illustrated by this particular example 
is generally true, and we are thus enabled to act upon 
the following postulate : 

All Galilean reference systems are equally suitable for 
the statement of general mechanical laws. 

This is what may be called the Mechanical Principle 
of Relativity. It will be observed that it is in reality 
not so much a principle of relativity as of correlativity , 
inasmuch as it points to the synthetic process of 
unifying different points of view rather than to the 
analytic operation of distinguishing between them. 

We have now to inquire into certain assumptions 
which were tacitly made in the discussion on the second 
law of motion which has just been given. 

It was assumed in the first place that the motion of 
0' relatively to made no difference in the value of 
the time t in the reckoning of either observer. evi- 
dently assumed that the clocks on the system of 0' 
registered exactly the same time as those on his own, 
and conversely. Each observer, in fact, assumed that 
he might, had he wished, have made use of the other 
observer's clocks for measuring time, or, in other words, 
that a second appeared to be exactly the same to 
both. 

In the second place, both of them assigned the same 
values to the velocities. For example, the velocity V 
which was gained by the particle was supposed by both 
to have the same value. It is true that, owing to the 
relative motion, the total velocity of P seemed different 



44 THE THEORY OF RELATIVITY 

to the two observers, but V was a gain in velocity, and 
was common to both. Now, it has been seen that a 
velocity is a comparison between length and time made 
in a special way, and since both observers believed their 
times to be the same, they must a'lso have believed their 
lengths to be the same. 

These assumptions, indeed, seem to be obvious 
common sense. For what difference, it may be asked, 
can the mere fact of movement make to the length of 
a rod ? Why should a yard measure carried by either 
observer appear any different to the other merely 
because it is moving relatively to him, or why should 
a clock appear to alter its rate merely because it is 
moving ? A railway passenger might as well expect 
his umbrella to shorten or lengthen, or his watch to 
gain or lose time when the train starts. 

Nevertheless, obvious as they may appear, it must 
be kept in mind that they are but assumptions, and like 
all other assumptions, they must be judged by their 
consequences. The question is not, whether they agree 
with common sense, but whether deductions which 
involve them agree with experiment. If they do not 
they must be reconsidered. 

As far as mechanics are concerned they have stood 
this test. The whole of physical astronomy, the pre- 
dictions of which have been verified so abundantly,* 
rests upon them, and so do the calculations of engineers 
and others concerning bodies in motion. They must, 
therefore, be very largely if not absolutely true. For 
the moment we shall assume their truth and proceed to 
derive from them geometrical formulae by which two 

* Excepting as already noticed in Chapter V. 



MECHANICAL PRINCIPLE OF RELATIVITY 45 



observers in relative motion can change over to each 
other's systems from their own formulae of transforma- 
tion, as mathematicians call them. 

Let P (Fig. 13) be a point referred to the same sets 
of axes which have already been considered in this 
chapter. Draw the ordinate PM, which is common to 
both systems, and the abscissae PN, PN', which 
coincide since O'x' is supposed to slide along Ox. Let 
0' ', as before, move with a velocity u relatively to 
along Ox, and suppose that and 0' coincide at zero 
time. Let the figure represent the state of affairs at 



N 



O 1 



M 



FIG. 13. 

the end of time t according to the reckoning of 0, and 
t r according to 0'. Let the co-ordinates of P, PN and 
PM in O's system, be (%, y), and let the co-ordinates 
PN' and PM in that of 0' be (*', /). Since both 
systems coincide at zero time 00', or NN', is equal 
to ut. Thus PN', according to 0, is PN - 00', or 
x ut. According to 0' ', PN' is x', and since both 
observers, in agreement with one of the suppositions 
above discussed, ascribe the same value to PN', we 
must have x' = x ut. PM is common to both 
observers, and we assume in a similar way that y = y'. 



46 THE THEORY OF RELATIVITY 

Had there been a third co-ordinate z, or z', we should 
similarly assume z = z'. 

Again, since by the Other supposition both observers 
ascribe the same value to the time, we shall have 



We thus have % x' + ut 

(or, x' = x ut) 



as a set of formulae which enable to express his 
observed lengths and times in terms of those of 0', 
and vice versa. They enable any statement made 
with reference to O's system to be converted into the 
corresponding statement with reference to that of 0', 
and conversely. The application of these formulae to 
the statements of general mechanical laws leaves the 
statements unaltered in form.* 



Summary. Mechanical laws cannot be expressed in 
similar form for systems moving with different accelera- 
tions. All Galilean systems are equivalent for the 
statement of mechanical laws. When we change over 
or transform a mechanical law from one Galilean system 
to another, it is assumed that each observer ascribes 
the same values to the other's lengths and times as to 
his own. Transformation formulae for correlating the 
observations of length and time of two observers under 
these two suppositions are determined. 

*The reader will observe that it is \hzform of the mathematical 
expressions which is preserved. Each observer states the facts in 
his own terms, e.g. x, y, z, t, or x\ y, z' t f, as the case may be. But 
all the statements agree as to the relationship in which these terms 
stand to one another. 



CHAPTER VII 
THE LORENTZ TRANSFORMATION 

THE correlation between different points of viewing 
mechanical phenomena was seen in the last chap- 
ter to depend, according to the classical mechanics, on 
two suppositions, which we may state as follows in 
somewhat more general terms : 

(1) The time interval between two events is indepen- 
dent of the condition of motion of the reference system. 

(2) The space interval between two points is inde- 
pendent of the condition of the motion of the reference 
system. 

From these were derived the set of formulae (i) of the 
last chapter, by which the suppositions could be applied 
to any given case of change of point of view. It was 
also seen that when so applied mechanical laws retained 
their form, so that these laws showed no preference for 
one system more than for another. These suppositions 
attribute an absolute character to space and time 
measurements which renders them independent of the 
motion of any particular observer. 

But when these suppositions are applied to the general 
laws of electro-magnetic phenomena it is found that a 
preference is shown. If the laws of the agencies which 
act in an electro-magnetic field are stated with reference 

47 



48 THE THEORY OF RELATIVITY 

to a system fixed with reference to the aether, which is 
the name given to the seat of those agencies or the 
medium in which they occur, and the transformation 
is applied to render the statement in terms of some other 
uniformly moving system, the velocity of this latter 
system does not cancel out as with mechanical laws, 
and the form of the statement is changed. As long 
as these suppositions are adhered to, a principle of 
relativity cannot be applied to electro-magnetics. 

In order to meet this situation, and in accordance 
with certain electro-magnetic facts, Lorentz proposed 
the following scheme of transformation in place of the 
scheme of the last chapter : 

x' = p( x - ut) 



Z'=Z 




(2) 



Where u, as before, is the relative velocity between 
two systems, and 

p = 

A/I 

c being the velocity of light in vacuo. It will be 
observed that x/i u*/c* is less than unity, so that 
j3, or i/^/i w 2 /c 2 , is greater than unity. When this 
transformation is applied to the mathematical state- 
ment of the laws of the electro-magnetic field, it is 
found that the velocity u cancels out as in the case of 
mechanical laws, and the form of the statement is 
preserved. 



THE LORENTZ TRANSFORMATION 49 

Comparing the first of equations (i) and (2) we see 
that the Lorentz transformation involves the abandon- 
ment of the supposition that observers' estimates of 
lengths in each other's systems are unaffected by 
movement. For, referring to Fig. 13, x ut is O's 
estimate of the length PN', while %' is that of 0'. 
The equation x' ft(x ut) means, therefore, that 
x' is greater than x ut, since fS is greater than unity. 
O's measurements of lengths in the system of 0' are 
less than those of 0' when these lengths are in the 
direction of motion. Electro-magnetic theory indi- 
cates the reality of this divergence. The equations 
y' = y and z 1 z mean that lengths at right angles 
to the direction of motion are unaffected. 

It might also be shown by comparison of the fourth 
equations of the two sets that the observers' estimates 
of times are likewise affected by the relative motion, 
so that the second supposition must also be abandoned, 
but as this point will be taken up later on, we shall not 
pause to consider it further at the present stage. We 
note, however, that the Lorentz transformation aban- 
dons the absolute character of space and time measure- 
ments and makes them dependent on the motion of 
each observer. 

Though the Lorentz transformation was supported 
by electro-magnetic theory, it could hardly be regarded 
as satisfactory. The object with which it was proposed 
was to preserve the form of certain physical equations 
in changing over from one reference system to another, 
and it could scarcely be looked on as more than a 
mathematical device invented ad hoc. Einstein, how- 
ever, showed that it followed at once without reference 
4 



50 THE THEORY OF RELATIVITY 

to electro-magnetic theory from a remarkable property 
of the measure of the velocity of light which we proceed 
to discuss in the next chapter. 



Summary. The suppositions of the absolute charac- 
ter of space and time measurements do not enable 
electro-magnetic laws to preserve their form on trans- 
formation. Lorentz proposed a scheme which effects 
this, but which involves the abandonment of the 
absolute character of the measurements. 



CHAPTER VIII 
THE VELOCITY OF LIGHT 

IT is proposed to examine in this chapter the two 
following postulates, and to draw from them a con- 
clusion regarding the velocity of light relative to an 
observer. The postulates are : 

(1) It is impossible for anyone to determine his own 
absolute unaccelerated velocity.* 

(2) The velocity of light in vacuo is independent of 
that of its source. 

We shall consider these in order. 

Every railway passenger must have experienced the 
difficulty of deciding whether his own train is moving, 
or an adjacent one, when the movement is slow. If 
he is moving and the movement is smooth and free 
from bumps or jolts, it is quite impossible to settle the 
matter by merely looking at the adjacent train. The 
passenger has to look out of the opposite window at 
some objects which he knows to be fixed relatively to 
the earth, such as buildings, or objects on the platform. 
In applying this test he cannot depend on any object 
on the train, that is, on his system ; he has to depend 

* This postulate, or a statement to the same effect, is given in 
many books as a definition of the principle of relativity (restricted)- 
The writer however has followed Einstein. See Preface. 

51 



52 THE THEORY OF RELATIVITY 

on something outside his system, to tell him whether 
he is at rest or in motion. But if the carriage should 
jolt, even once, he recognizes his movement imme- 
diately. Now, a jolt or bump is a change in his motion, 
that is to say, an acceleration ; consequently it is 
only when there is no acceleration, or when the accelera- 
tion is too slight for notice, that he has to look for 
external evidence. Failing noticeable acceleration there 
is nothing inside his compartment to tell him, since 
everything shares his state of rest .or movement. But 
even if he has decided that he is actually moving, all 
he knows is that he is moving relatively to the earth. 
For all he knows the movement of the spot where the 
train is situated, due to the rotation and translation 
of the earth and the bodily movement of the solar 
system, may be such as exactly to cancel the motion 
of the train relatively to the earth, so that in point of 
fact he is at rest. 

The same failure to discover absolute motion attends 
all mechanical experiments. As we have seen, mechani- 
cal laws have no preference for one uniformly moving 
system over another, and no distinction can be drawn 
between them. The fact is that there is no body of 
reference which is known to be fixed, and to which 
reference can therefore be made to determine the state 
of rest or movement of any other body. Without such 
a fixed body absolute motion is an unmeaning expres- 
sion ; only relative motion is determinate. 

It was at one time thought that such a reference body 
could be found in the aether of space, using the word 
" aether " for the vehicle of transmission of light waves 
and other electro-magnetic radiations, such as heat or 



THE VELOCITY OF LIGHT 53 

wireless rays, without assigning any other properties 
to it. Certain experiments seemed to indicate that 
the aether was fixed, or, at all events, possessed only 
such motion as was shared by the whole visible universe, 
and should therefore be ignored. It was, therefore, 
hoped that electro-magnetic experiments including 
experiments with light, which is known to be of electro- 
magnetic origin would enable the earth's motion of 
translation to be determined with reference to it. But 
all these experiments failed.* No movement of any 
kind could be detected. It is, however, a long step 
from what experiment shows, namely, that no motion 
has been detected, to the statement of the postulate that 
no motion can be detected. It would, of course, be 
wholly unjustifiable if there were any reason to think 
that the experiments were of so clumsy a nature as to 
fail to detect a movement which really existed, or if 
there were any suspicion that all possible means had 
not been exhausted. So far from the experiments being 
clumsy, they were of so refined a character as to render 
possible the detection one-tenth or less of the expected 
result. The earth travels in its orbit at the rate of 
about 30 kilometres a second, and as the experiments 
were repeated at all times and seasons, the locality 
where they were conducted must have had this velocity 

* The experiments were : 

Michelson-Morley, "American Journal of Science," 3rd series, 
Vol. 34 (1887), pp. 333-345 5 also, " Phil. Mag.," Vol. 24, 5th series, 
Dec., 1887; Morley and Miller, "Phil. Mag.," Vol. 9, 6th series, 
May, 1905; Trouton and Noble, "Proceedings R.S.," Vol. 72 
(1903), p. 132; also "Phil. Trans.," Vol. 202 (1903), p. 165; 
Trouton and Rankine, " Proceedings R.S.," Vol. 80 (1907 and 1908), 
p. 420. 



54 THE THEORY OF RELATIVITY 

at least at some time. The experiments, however, 
would have detected a velocity of 3 kilometres per 
second. Moreover, mechanical and electro-magnetic 
means having failed, there is absolutely no other known 
agency available for experiment. The conclusion, 
therefore, seems unavoidable that the motion is un- 
detectable, and we are, therefore, justified in adopting 
the postulate. What interests us in particular for 
present purposes is the fact that motion relative to the 
medium which transmits light is undetectable. 

We now consider the second postulate, which states 
that the velocity of light is independent of that of its 
source. The meaning of this postulate can be made 
clear by an example. If a gun, whose muzzle velocity 
is 2000 feet per second, is fired in any direction from an 
armoured train at rest, the velocity of the shell will in 
all cases be the same. But if the train is moving at 
the rate of 15 miles an hour, or 22 feet per second, the 
velocity of the shell, if the gun is fired directly forward 
without elevation, will be 2022 feet per second, or if 
fired directly backward, 1978 feet per second. The 
velocity of the train affects the velocity of the shell to 
the extent of its own velocity of 22 feet per second 
plus or minus. The case of a ship at sea is wholly 
different. The waves due to the motion of the ship 
recede from it always at the same velocity indepen- 
dently of the speed of the ship. The only difference 
which the speed of the ship makes is in the sizes and 
lengths of the waves. So also with sound waves ; the 
velocity of a source of sound affects the lengths of the 
waves, but not their velocity when once started. In 
fact, the independence of the velocity of a wave and 



THE VELOCITY OF LIGHT 55 

its source is characteristic of wave motion generally. 
Now light, according to the wave theory, consists of 
waves, not in water or air, but in the aether, and light 
waves have the same property as others ; their speed 
is always the same. The postulate is, therefore, a direct 
consequence of the wave theory of light, but it has been 
proved independently of any theory of light by observa- 
tions on the fixed stars.* 

The conclusion from these postulates is obvious. If 
the velocity of light in its medium is an absolute con- 
stant, and the observer cannot perceive his motion 
through that medium, it necessarily follows that the 
velocity of light relative to him must always appear to 
him to be the same and equal to its constant velocity 
in its medium. Or, we may put the matter in the form 
of a reductio ad absurdum thus : If the second postulate 
is accepted, and if in addition an observer could perceive 
any difference between the absolute velocity of light 
and his velocity relative to light, that difference would 
enable him to measure his own velocity in the medium, 
which is contrary to the first postulate. 

As many people find considerable difficulty in accept- 
ing the proposition which has just been proved, some 
indeed considering it so preposterous as to amount to 
a reductio ad absurdum of the whole subject of relativity, 
it is necessary to examine its terms. A form in which 
it is sometimes stated, namely, that the velocity of 
light is the same for all observers, is certainly open to 
serious misconstruction. The statement really means 
that the velocity of light relatively to each observer 

* Einstein, "Relativity," p. 17. Also "The Principle of Rela- 
tivity," Saha and Bose (Calcutta University), pp. 172 et seq. 



56 THE THEORY OF RELATIVITY 

always appears the same to him, and equal to its con- 
stant velocity in its medium. It does not mean that 
the velocity of light relative to each observer appears 
the same to other observers. Thus, consider a light beam 
AB having its source at A, and let there be two ob- 
servers 0, 0' , the former of whom may be supposed to 
be stationary relatively to A , while the latter moves in 
the direction of the beam with a velocity u relatively 
to the former observer. The velocity of light in vacuo 
being 300,000 kilometres per second, makes out the 
velocity relative to himself to be 300,000 kilometres per 
second, and 0' also makes out the velocity relative to 
himself to be the same, but computes the velocity of 

A B 



O 1 



FIG. 14. 

light relative to 0' to be 300,000 kilometres per second 
minus u t and 0' computes the velocity of light relative 
to to be 300,000 kilometres per second plus u. The 
reason why and 0' make out the velocity to be the 
same, each relative to himself, is, as we shall presently 
see, because their relative motion affects their measure- 
ments of lengths and time. These measurements adjust 
themselves automatically in such a way as to give the 
same numerical value to the relative velocity in both 
their cases. 

But, however we may regard it, the proposition is 
sufficiently strange and difficult to grasp at first. It 
obviously makes the velocity of light unique amongst 
all other velocities ; the velocity of nothing else appears 
to every observer to be the same relative to himself. 



THE VELOCITY OF LIGHT 57 

It differs from all other wave motions because in every 
case, except that of light, the observer is able to recog- 
nize his movement through the medium in which the 
waves occur. 

An example will illustrate the remarkable physical 
consequences of the proposition. Suppose two ships to 
pass at sea, going in opposite directions, and suppose 
that when they are abreast a splash is made in the sea 
between them. The waves spread out in all directions, 
and the ships go on, as it were, in pursuit, leaving behind 
them the place where the splash occurred. Neither 
ship is in any doubt about this, they are aware that 
they do not remain at the centre of the disturbance. 
But now suppose two observers to pass one another in 
space, and a light signal to be flashed between them. 
It might be supposed that the same thing would happen 
as in the case of the ships, but this is not so. As far 
as his observation can tell him, each observer thinks 
that the centre of the disturbance remains with him. 

It is easily seen that the postulates imply that 
velocity of light must appear the same in all directions 
to any observer. 



Summary. Two postulates are stated and explained. 
The essential feature of the first is that no observer can 
detect his motion through the medium which serves 
as the vehicle for the transmission of light waves and 
other electro-magnetic radiations. The second is that 
the velocity of light through this medium is independent 
of that of its source. It is thence deduced that the 
velocity of light has the same measured value relative 
to every observer. 



CHAPTER IX 
THE RESTRICTED PRINCIPLE OF RELATIVITY 

WE shall now show how the results of the last 
chapter affect the measurements of length of 
two observers moving with uniform relative velocity, 
and thence deduce the Lorentz transformation.* We 
shall then state and explain what is known as the Re- 
stricted Principle of Relativity. 



o 1 



FIG, 15. 

* The discussion which follows is not regarded by the writer as 
the simplest from a mathematical point of view, but the direct 
method involves more algebra than he is allowing himself. For 
this latter method, reference may be made to Einstein's book, 
"Relativity," Appendix I, or to "The Electron Theory of Matter" 
(O. W. Richardson), pp. 297-300. The method in the text was in 
part suggested by a paper by R. D. Carmichael in the "American 
Physical Review," Vol. XXXV, No. 3. Carmichael, however, 
applies the method to time, not length measurements. 

58 



THE RESTRICTED PRINCIPLE OF RELATIVITY 59 

Let us imagine two observers, and 0' (Fig. 15), in 
uniform relative motion with velocity u, but, as we 
have seen, neither can tell whether he is moving or not. 
0' provides himself with an apparatus consisting of two 
equal and mutually perpendicular arms O'M, 0'M lt at 
the ends of which are mirrors M and M it each perpen- 
dicular to its arm. The arm O'M lies in the direction 
of relative motion. 0' sends out light signals simultane- 
ously along each arm from their intersection 0' . These 
signals are reflected back, and since the arms are of equal 
length and the velocity of light the same in every direc- 
tion, the signals return to their starting point in the same 
time and reach it simultaneously. Or, if the reader 
prefers, 0' adjusts the lengths of the arms so that the 
signals return to him simultaneously, and he judges 
them to be equal. It does not matter how the adjust- 
ments are made ; the essential point is that 0' judges 
the arms to be of equal length, and the signals to take 
the same time over their journeys. All this takes place 
whether 0' is moving or not. If his motion made any 
difference to him that difference would enable him to 
detect it, and this, we have agreed, is impossible. 

Now is looking on at these proceedings, and how 
does he regard them ? He, like 0', is unable to say 
whether he is moving or not. All he knows is that 0' ', 
with all his instruments, is moving past him with a 
velocity u, as in Fig. 16. He sees 0' sending signals 
along the arms, and he knows that 0' receives them 
simultaneously. He may be supposed to know this 
either because he can see the instruments belonging to 
0' ' , or because 0' may have told him so. What he is 
not supposed to know is that 0' thinks that the two 



6o 



THE THEORY OF RELATIVITY 



arms are of equal length, and he proceeds to compute 
their relative lengths from the fact that the signals 
perform their several journeys in the same time. It 
does not matter whether his reckoning of the time is 
the same as that of 0' ', since it will be found to disappear 
from the equations. Provisionally, calls the length 
of the arm in the direction of motion X, and the other 
arm /, and he proceeds to relate / and X by determining 
the times which the signals take over their journeys 
and equating them as follows : 



M 




0' 




FIG. 16 

i. THE SIGNAL IN THE DIRECTION OF MOTION 

Let tj_ be the time the signal takes to go from the 
starting point 0' up to the mirror M, which, by the 
time the signal reaches it, will have moved according 
to O's reckoning from the place M, where it was when 
the signal left 0' to some point M'. We shall thus 
have O'M' = ct\ t where c is the velocity of light. Now 
MM' is equal to the distance which both 0' and the 
mirror have moved in the same time t L with their 
common velocity u, and therefore MM' = ut^ Also, 
O'M' = O'M - MM' = X - MM', and therefore 



THE RESTRICTED PRINCIPLE OF RELATIVITY 61 

or, /, = ; 

c -f u 

Meanwhile 0' has, according to 0, moved to 0" ', where 
O'O" = *L 

We have now to consider the return of the signal to 
0'", which is the position of 0' when the signal gets 
back to him. If t 2 be the time taken, we see that the 
light covers the distance M'O'" in the time t z , and thus 
M'O'" = ct, L . In the same way as before, we have 
0"0'" = uk, and M'O" = X. 

Thus ci 2 = X -f- ut z 

X 

or, L = . 

c u 

Now T, the whole time for the out and home journeys, 
is ^ + ^, that is, 

T ^ 4. ^ 



by the rule in algebra for the addition of fractions. 

2. THE SIGNAL AT RIGHT ANGLES TO THE 
DIRECTION OF MOTION 

In this case the signal describes the two equal sides 
of the isosceles triangle O'M/O'" in the time T that 
is, each side is described in the time T/2 ; and since 
the velocity of light is the same in all directions, it is 
still c as before. Thus 

O'M/ = cT/2 

where M/ is the position of the mirror M t at the end 
of the time TJ2. During this time 0' has moved with 



62 THE THEORY OF RELATIVITY 

velocity u to P, where P is the foot of the perpendicular 
from M/ on the direction of motion of 0'. O'P, 
therefore, is uT/2. Now MI P is what calls /, and 
in the right angled triangle O'M/P 

O'M/ 2 = O'P 2 + M/P 2 
by the theorem of Pythagoras, 






4 4 
that is, r= , 2/ 




We therefore have two expressions for T which we can 
equate, and thus find 

2\C 



or, 



Thus, X (the length of the arm in the direction of 
motion), which according to the reckoning of 0' was 
equal to / (the length of the arm at right angles to the 

VI0P 
i ^' 

1 ~ 2 

That is, computes it to be shorter, since Ji :, 

is less than unity. This it will be remembered is 
exactly the contraction assumed in the equations of 
Lorentz. 

Now suppose 0' to be the spectator and the experi- 
menter. The circumstances are altered in no material 
respect. All that 0' knows about the motion is that 



THE RESTRICTED PRINCIPLE OF RELALIVITY 63 

is passing him, and thus since both attribute the 
same velocity to light, a length measured in the direction 
of motion which supposes to be /, is I^/T i\c l 
according to 0'. Each attributes a contraction to the 
other's lengths when these are measured in the direction 
of motion, and there is no means of deciding which 
determination is to be preferred.' 

Since u is small in ordinary cases compared with c, 
which is 300,000 kilometres per second, it will be seen 

f? 
i 2 , though less than unity, does 

not differ much from it, so that the contraction is small. 
Though the velocity of the earth in its orbit is 30 kilo- 
metres per second, the contraction in its diameter as 
seen from the sun would only amount to about 2% inches. 
The remarkable fact, however, is not the magnitude of 
the effect, whether small or great, but its occurrence. 
To produce a substantial contraction, an enormous 
relative velocity would be required. Let us find what 
relative velocity would produce a contraction of one- 
half. To do this we have only to put Ji. ? equal 

\ C" 

to one-half, and work out the value of u. 

Thus - = 

tfl i 
or i = - This gives us 



u 2 3 u x/3~ 7 

^r = -, or - = -^ = | approximately. 



Taking c as 186,000 miles per second, this gives 
u =* 161,000 miles per second approximately. 



64 THE THEORY OF RELATIVITY 

It should be observed that the contraction is attri- 
buted to a body in the direction of motion only. There 
is no evidence that it takes place in any other direction, 
and its absence is therefore presumed. This is, of 
course, an assumption, just as the absence of all altera- 
tion whatever was an assumption in the classical 
mechanics. The present assumption is, however, so 
far uncontradicted by any experiment, and there is no 
known phenomenon which suggests anything to the 
contrary, or which it would help us to explain. If in 
the future anything should come to light suggesting 
that movement affects lengths perpendicular to its 
direction the matter would, of course, have to be 
reconsidered, but until then such an assumption would 
be gratuitous. 

We have now to deduce the Lorentz transformation. 
Referring to Fig. 17, let two observers, and 0', move 
relatively to one another with uniform velocity, u, so 
that the #-axes of both coincide. For simplicity we 
shall, in the first instance, consider two spatial dimen- 
sions only that is to say, all the phenomena will take 
place, and the measurements will be supposed to be 
made, in the plane of the paper. The extension to 
three spatial dimensions is easy. 

Let P be any point fixed in the system of 0', Draw 
PM perpendicular to Ox, or O'x', and PN'N perpen- 
dicular to O'y' , or Oy, to meet these two lines in N f 
and N. Let PM = y, or /, and PN' = x', while 
PN = x. Let us suppose that 0' coincides with at 
zero time, and that the system of 0' has come into the 
position shown in the figure after a lapse of time t, 
according to the reckoning of 0, and t' , according to 



THE RESTRICTED PRINCIPLE OF RELATIVITY 65 



the reckoning of 0'. Then 00' = ut, and PN', or 
PN 00', according to O's reckoning is x ut. This 
is the distance which 0' calls x 1 '. But by the result 
which has just been obtained this distance appears to 

- 

1 ~~' 
that is 



/ u z 

X \I $ = X Ut. 


N 




N p 





0' 


MX' 



ar = 




Thus 



or x' = p(* - *) . . . (3) 

where ft is written for ^ = = for brevity. 



We may relate # and #' in terms of t' in the same way. 
PN is equal to PN' + NN'. PN' as before is x', 
according to the reckoning of 0', and NN' is ut'. Thus 
the distance PN is #' -f ut' in the reckoning of 0'. 
This is the distance which calls #, and it therefore 
appears to 0' to be x*/i u' 2 /c*. We thus get 

Xf ji u ' 2 Jc? = x' + ut', 
or x = )9(^' + frf'). . . . (4) 



66 THE THEORY OF RELATIVITY 

By means of (3) and (4) we can relate t and t' in 
terms of either x or x r . Substituting for %' in (4) the 
value /3(x ut) given by (3) we get 

x = p{p(x - ut) + ut'\ 
= P*x - put + put'. 
Thus put' = x(i - 2 ) + 



Thus put' = p 2 ut - U ~ p, 

or, dividing out by @u, 

'-*(-?). ... (5) 
In a similar way we might have obtained 



by substituting for x in (3) the value /3(x' + ut') given 

by (4)- 

Since the relative movement does not affect lengths 
at right angles to it, we have 

3/=y. 

To introduce a third dimension, all we have to do is 
to move P, N' and N out of the plane of the paper 
towards the reader through a distance z or z r in the 
obvious way shown in Fig. 18. z, like j>, is perpendicular 
to the direction of motion, and therefore 

z= z'. 
x 



THE RESTRICTED PRINCIPLE OF RELATIVITY 67 



Collecting all these results we have 
x' = &(x - ut) 



and 



. (6) 



x = p(x' + ut') 

y=y' 



ux' 



(7) 



N 



FIG. 18. 

We are thus brought by perfectly general considera- 
tions back to the equations of the Lorentz transforma- 
tion, which are shown in this manner to be independent 
of any electro-magnetic phenomena, and to be conse- 
quences of relative motion pure and simple. 

We have seen that the observers ascribe a contraction 
to each other's lengths ; let us now see how they regard 
each other's times. Consider first a clock at 0' on the 



68 THE THEORY OF RELATIVITY 

system of 0', and let us see how regards it. Since 
the clock is at the origin 0', we have %' = o. Thus 
from the fourth equation of (7) above we have 

t = /3t f . 

Now t' is the time indicated on this clock according 
to 0', and t is the time indicated on the same clock 
according to 0. ft is greater than unity, so that in the 
opinion of a longer time has elapsed since zero time 
than in the opinion of 0'. Thus all movements on the 
system of 0' appear to to be more lethargic than to 
0'. If, for example, an airplane pilot passing at the 
rate of 161,000 miles per second extends his arm in the 
direction of motion, the arm when extended appears 
only half as long to as to 0' ', but the process of 
extension appears to take twice as long to as to 0'. 

This opinion is reciprocal. Consider a clock at on 
the system of 0, and put x = o in the fourth equation 
of (6) above. We thus get t' = fit. By the same 
argument as before we see that 0' thinks O's clock 
goes slow, and thus if stretches out his arm in the 
direction of motion the airplane pilot thinks that the 
arm has contracted to half its length, and that the 
extension takes twice as long, compared with O's reckon- 
ing. and 0' each think that the lengths of the other 
have contracted and their times lengthened. 

By the use of equations (6) or (7) we can transform 
or change over from the point of view of one observer 
to that of another, while in so doing, as was seen in 
Chapter VII, electro-magnetic laws preserve their form, 
the observers being supposed to use Galilean frames of 
reference. We are thus able to say that 



THE RESTRICTED PRINCIPLE OF RELATIVITY 69 

All Galilean frames of reference are equally suitable for 
the statement of general physical laws. 

This is called the Restricted Principle of Relativity. 
It is called " restricted " because its application is 
confined by the terms of the definition of a Galilean 
frame of reference to unaccelerated systems.* 

Finally, the reader should note that the equations 
(6) and (7) are merely the embodiment in form suitable 
for transformation of the different estimates which 
observers make of each other's lengths and times, just 
as equations (i), Chapter VI, embody. the suppositions 
that their estimates are the same. 



Summary. The fact that the velocity of light relative 
to every observer is the same causes observers to ascribe 
contractions to lengths on each other's systems 
measured in the direction of motion. Lengths per- 
pendicular to this direction are unaltered. The Lorentz 
transformations can be deduced from this fact. Ob- 
servers also think each other's times are longer than 
their own. These different estimates of length and 
time render possible the statement of a principle of 
relativity which includes all physical laws, and they are 
embodied in the formulae which enable observers to 
change over from one point of view to another. 

* It is also called the "special" principle for the same reason. 



CHAPTER X 

SOME SPECIAL FEATURES OF THE RESTRICTED 
THEORY OF RELATIVITY 

THE purpose of this book being to explain the 
principles upon which differently circumstanced 
observers can state their facts in general form so that 
they can all tell the same story, the last chapter strictly 
speaking brings us to the end of what has to be said 
on the subject of the Restricted Principle of Relativity. 
The whole theory arising from the application of this 
principle to physical laws generally is beyond our 
purpose. Indeed, the development of the General or 
Gravitational Theory of Relativity, which will be dealt 
with in due course, has robbed the restricted theory of 
much of the interest which it originally possessed, 
excepting in the field of electro-magnetics. The General 
Theory has no logical dependence upon the restricted 
theory, as will be seen ; but the restricted theory is an 
almost indispensable introduction to it. A geometrical 
development to which it leads is, in fact, essential to 
the general theory. This development will form the 
subject ot the next chapter, but meanwhile we give, 
by way of digression, some results of such interest or 
importance as to require reference. 

70 



THE RESTRICTED THEORY OF RELATIVITY 71 

i. MECHANICS 

The Newtonian laws of mechanics, having been 
originally stated with reference to Galilean systems 
under the suppositions of unalterable lengths and 
times as between differently circumstanced observers, 
naturally require modification if stated in accordance 
with the suppositions implied in the transformation of 
Lorentz. It is impossible to do more than refer to this 
matter here, but something further of a general charac- 
ter will be said on the subject in the next chapter. To 
give even a summary of the particulars would mean a 
treatise on mechanics, and in addition to this, since 
mechanics and electro-magnetics are now so closely 
connected, a greater knowledge of this latter subject is 
required than all the readers of this book can be 
assumed to possess.* 



2. SlMULTANEOUSNESS OF EVENTS AND RATES OF 

CLOCKS. 

Let two separate events take place at points (x 1 y l zj 
(x 2 y 2 z 2 ) and times ^ and t. 2 in O's reckoning, and 
( x i y\ z i) ( X 2 f yz Z 2 f ) an d times ^' and t 2 ' in the reckoning 
of 0' '. Then by the fourth equation of (6), Chapter IX, 
we have 

* Those who wish to follow up this matter will find it developed 
in such books as the following: "The Theory of Relativity," by 
R. D. Carmichael (Chapman and Hall) ; " The Principle of Rela- 
tivity," by E. Cunningham (Cambridge University Press) ; " The 
Theory of Relativity," by L. Siberstein (Macmillan). The first of 
these is the simplest. 



72 THE THEORY OF RELATIVITY 



Subtracting the first of these equations from the second 
we get 



2 . 

We can draw two conclusions from this : 

(1) If the events are simultaneous according to 0, 

t't = *i> or * 2 t\ = o. 
Thus V - V = - /^ 2 (* 2 - *0- 

Now, if the events are also simultaneous according to 
0', t* must be equal to //. But this cannot be the case 
unless x z = x^ By using the fourth equation of (7), 
Chapter IX, we can show in the same way that #/ 
and %2 must be equal. The ^-distances of the two 
events must, therefore, be the same to both observers 
that is to say, the events must occur at some place 
situated in a plane perpendicular to Ox. 

(2) Suppose the events to occur at the same place. 
Then x l = x 2 and 

v-*i' = /3(* 2 -y. 

Now let the events be two successive beats of a clock 
which beats seconds. Then t 2 ^ is a second according 
to O's reckoning, and t. 2 f t/ is a second according to 
that of 0'. Now ft is always greater than unity, and 
therefore a clock beating seconds on the system of 0' 
appears to to beat more slowly than one on his own 



THE RESTRICTED THEORY OF RELATIVITY 73 

that is, the clocks on the system of 0' appear to to 
go at a slower rate than those on his own. This dis- 
cussion amounts to the same thing as that given at the 
end of Chapter IX. 

If 0' were to move with the velocity of light 2 

V 1 -a 
C 2 

would become infinite. Thus an observer at would 
see the pendulum of the clock at 0' at the commence- 
ment of a swing, but he would never see it reach the 
other end. The clock would appear to to have 
stopped altogether. As it is not desirable to introduce 
considerations of the velocity of sound, we have pre- 



M 



C M' D 

FIG. 19. 

f erred the above form of statement to saying " would 
hear the first beat of the clock at 0', but he would 
never hear the second beat ". But it comes to the 
same thing. 

We give an illustration, due to Einstein, to show that 
events which appear simultaneous to one observer are 
not necessarily so to a second observer moving with a 
relative velocity. 

A B (Fig. 19) is a train moving in the direction of the 
arrow. When A and B are opposite points C and D 
on the permanent way, flashes of lightning occur at 
C and Z), and will be judged to be simultaneous by an 
observer standing by the line at M', the point midway 
between C and D. Let M be the middle point of the 



74 



THE THEORY OF RELATIVITY 



train, which is opposite M' when the flashes occur. 
Now, an observer at M is travelling towards D and 
away from C. Consequently he is meeting the light 
coming from D, and moving away from the light coming 
from C. The flashes will, therefore, not appear to be 
simultaneous to him. It is hardly necessary to state 
that since either observer can consider himself at rest, 
the above effects are reciprocal. For example, each 
observer thinks that the other observer's clocks go 
slower than his own. 



P 1 



o ( 



FIG. 20. 

3. VELOCITY AND ACCELERATION 

(i) Velocity in the Direction of Motion, i.e., parallel to 
the x-axis 

Let V be the velocity of a point P (Fig. 20) rela- 
tively to 0' in the direction Ox (or Ox'}. Let x^ and 
x% be the ^-distances of P at times 2/ and t 2 ' respectively. 
Then P moves relatively to 0' through a distance PP', 
or x 2 ' #/ in a time t z ' t^, so that 



V-tt" 



THE RESTRICTED THEORY OF RELATIVITY 75 

We have to find the velocity of P as it appears to 0. 
If we call this velocity u^ we shall have 



_ 



where the #'s and tf's correspond in O's reckoning to the 
x"s and t"s given above in that of 0' . 

Substituting for x lt x. 2t t lt 1 2 from the first and fourth 
of equations (7), Chapter IX, we have 



Ui = - - t j 

The P's all cancel, and we have 



Dividing the numerator and denominator by t. 2 f t\ 
we get 







M - * 

' ' ~ 



This result states the addition theorem for velocities 
according to the relativist view. In the classical 
mechanics, if a point is moving with a velocity , and 
a second point is moving relatively to the first with a 
velocity %' in the same direction, the velocity of this 
second point is u + %' Thus, if a train is moving at 
the rate of 40 miles per hour, or 66 feet per second, 
and a passenger walks along the corridor at the rate 



76 THE THEORY OF RELATIVITY 

of 3 feet per second in the same direction, his velocity 
relatively to the ground, according to the classical view, 
is 69 feet per second. According to the relativist view 
it is 

^| feet per second, 



c 2 

where c is the velocity of light expressed in feet per 
second, which is somewhat less. 

It is interesting to see what the result is when %' is 
the velocity of light that is to say, if a point is moving 
with a velocity u relatively to an observer who considers 
himself fixed, and a light beam is sent out in the same 
direction from the relatively moving point, what is its 
velocity with reference to the " fixed " observer ? 
Putting MI = c, we have 

u + c 



Thus the velocity relative to the fixed observer is 
also c. This is not a new result, as it was the basis of 
that of the last chapter, from which the present results 
have been derived, but it is interesting to note that 
the present result is consistent with the previous one. 

Lest the reader should think that we are trying to 
bewilder him with paradoxes, it may be well to remind 
him that we are speaking all the time of physical 
measurements. It is on points such as the present that 
a person is apt to lose himself by unconsciously import- 
ing metaphysical ideas of extension and duration. 
What the statement which has just been made means. 



THE RESTRICTED THEORY OF RELATIVITY 77 

is that if anyone actually measures the velocity of light 
it will always relatively to himself figure out to the 
same number. Similarly with the statement that no 
velocity can exceed that of light, which is also true in 
the physical sense. This may be proved in many ways, 
the present amongst them. We have just tried to add 
a velocity u to that of light, and we find that the result 
comes out to be the velocity of light over again. What 
we mean, is that if there is such a velocity there is no 
way of recognizing it, for no means exist for measuring 
it. 

(2) Velocity at Right Angles to the Direction of Motion 

Let Vi be the velocity of P relatively to 0' in the 
direction Oy (or Oy'}, and let y x ' and y 2 ' be the y-dis- 
tances at the times t^ and t 2 '. Let v v y lt and y 2 corre- 
spond in O's reckoning to v^, y> and y z ' respectively. 
Then 

y* - yi y-i - y\ 



i 

' 



,' 



/3 u * #1 ft T , w/i 

1 T ~ 2 -p / 1 " ~3 

C/ t/2 - *i v 

Similarly, if w^ and w^ are the relative velocities in the 
direction Oz 

I Wi' 



78 THE THEORY OF RELATIVITY 

Here it is to be noticed that although lengths per- 
pendicular to .the direction of motion appear the same 
to both observers, velocities perpendicular to the 
direction of motion do not. We see, also, that the 
parallelogram of velocities does not hold good in the 
classical form. According to the classical view, if a 
point is moving with a velocity u relatively to an 
observer 0, and another point is moving in a direction 
at right angles with a velocity v ' relatively to the first 
point, the resultant velocity is represented by the 




diagonal AB (Fig. 21) of a rectangle, the sides of which 
are proportional to u and v^ that is, the resultant is 



+ "i". 

According to the relativist view the resultant is AB' , 
and is equal to 



relatively to 0. 

We might pursue the same line of argument with 
respect to accelerations, but this matter is not of 



THE RESTRICTED THEORY OF RELATIVITY 79 

immediate interest. The results, as might be expected, 
are of the same form as for velocities. 

4. MASS 

According to the classical view, mass has always been 
held to be the same for the same body under all con- 
ditions of motion. We shall now inquire whether 
relative motion will affect its measure in like manner 
to the measures of length and time.* 

U 





As a preliminary, let us consider the following case. 
Referring to Fig. 22, let two smooth, perfectly elastic 
spheres, A of mass m and B of mass m', moving in 
straight lines with velocities U and V respectively in 
any direction, collide. The size of the spheres is 
immaterial. By a process which is well known, and 
which the reader may take for granted, we may replace 

* The substance of this section is taken from Carmichael's book 
"The Theory of Relativity " (p. 49), cited above. See also Lewis 
and Tolman, "Phil. Mag.," 18, pp. 510-523 



8o THE THEORY OF RELATIVITY 

the velocity U by two others, v and u, v perpendicular 
to the tangent CD to both spheres at the point of 
collision, and u parallel to it. The velocity V may be 
replaced by two velocities, v' and u', in a similar way. 
Now the laws of mechanics tell us that the collision does 
not affect the velocities u, u 1 ', but in the direction 
perpendicular to CD the velocities are modified, though 
in such a way that after collision the total amount of 
momentum in this direction is the same as before. 
The spheres still preserve their original momenta due 
to the velocities u and u f , so that their total momenta 
are in some inclined directions such as are shown by 
dotted lines ; this, however, is immaterial for present 
purposes, the point being that whatever exchange of 
momentum takes place in directions perpendicular to 
CD, its total amount is unaltered. All this is merely 
a particular case of the general rule that the quantity 
of motion that is, the momentum of any system of 
bodies is unaltered by collisions or other actions between 
the bodies. If the momenta of the two bodies in this 
perpendicular direction are numerically the same before 
collision the exchange will not affect the magnitude of 
either ; it will simply reverse its direction.* 

Referring to Fig. 23, suppose two observers to 
be provided with spheres of equal mass and size. 
We will imagine the observers to have met and com- 
pared the spheres, so that when the comparison is being 
made no question of relative velocity arises. We next 
suppose the observers to separate and move off some- 

* See any book on dynamics, e.g., Tait's "Dynamics," pp. 198 
and 199. 



THE RESTRICTED THEORY OF RELATIVITY 81 

where into space, and to acquire a velocity u relative 
to each other. One of them, 0, may consider himself 
fixed, and regard 0' as moving past him with the 
velocity u. Each projects his sphere at right angles 
to the direction of motion and with the same velocity v, 
each according to his own reckoning, in such a way that 
when and 0' are directly opposite one another the 
spheres collide exactly midway at A. This means of 



0, 



0' 






FIG. 23. 

course that O f will have to project his sphere at some 
place Oi slightly before he comes opposite to 0. The 
sphere belonging to 0' will have, in addition to v, the 
velocity u in the direction of motion. It will preserve 
this velocity u after collision, since the spheres are 
smooth, and it will return to 0', whom it meets when 
he has reached a point 2 ', such that 0'0 2 f = O'O/, so 
that the momentum due to this velocity is the same as 
before. The sphere simply retains it. The spheres 
6 



82 THE THEORY OF RELATIVITY 

interchange the other velocities, but since these are 
the same, the exchange makes no difference, and each 
returns to the thrower with the same velocity v. Now 
what does infer from the fact that his sphere returns 
to him with the same velocity as that with which he 
projected it ? He infers that the other sphere must 
have had the same momentum as his own. If in his 
reckoning the mass of his own sphere is m, and its 
velocity v, and the mass of the other sphere is m and 
its velocity v lt he infers that mv = m^. Now the 
distances OA, O'A are the same to both observers, 
since relative velocity makes no difference to lengths 
perpendicular to the direction of motion. But the 
time which ascribes to the trajectory of the sphere 
belonging to 0' appears to him to be /3 times longer 
than his own corresponding time, where /? as before 
is equal to i/x/ 1 w 2 /c 2 , and therefore the sphere 
belonging to 0' seems to him to move more slowly 
than his own, and to have a velocity v//3. Thus v 1 v/ft 

and mv = m^/fi 

or M! mfi .. = . 

Vi - u*/c* 

Thus the sphere belonging to 0' appears to have 
increased in mass since the time when the observers 
made their comparison under the same conditions. 

As before, this opinion is mutual. 0' thinks that 
O's sphere has increased in mass compared with his 
own, though when they compared the two spheres 
together the masses were the same.* 

* The foregoing discussion relates to what is called " transverse " 
mass ; that is, mass measured transversely to the direction of relative 



THE RESTRICTED THEORY OF RELATIVITY 83 

This result is of considerable importance, as the 
negatively electrified particles called electrons which 
are ejected from radio-active substances exhibit changes 
in mass. Since the velocities of the electrons in such 
cases may be of the order of that of light, these changes 
may become observable. 



Summary. (i) Mechanical laws require restatement, 
in view of the suppositions of variable lengths and times 
which underlie the Lorentz transformation. 

(2) Events which appear simultaneous to one of two 
observers in relative motion are not generally simul- 
taneous to the other. 

(3) Under the above circumstances each observer 
thinks that the other observer's clocks go slower than 
his own. 

(4) Velocities and accelerations in the same direction 
cannot be compounded by the simple process of adding 
them or subtracting one from the other. 

(5) The parallelogram of velocities (and of accelera- 
tions) does not apply in the form stated in the classical 
mechanics. 

(6) Mass appears to increase with velocity. 

motion. Mass measured in the direction of this motion is called 
" longitudinal " mass. It is subject to yet another change, but it 
is of no interest or importance. As it can only show itself in the 
direction of motion, and an enormous velocity would be required to 
make it measurable, it cannot be made the subject of experiment. 



CHAPTER XI 
THE FOUR-DIMENSIONAL CONTINUUM 

WE shall discuss in this chapter the geometrical 
implications of the Lorentz transformation. We 
shall first consider the case of two dimensions of space 
only. 

Let P and Q (Fig. 24) be any two points whose 




M 



N 



M 



JC 



FIG. 24. 

positions are determined by reference to some rect- 
angular frame, Ox, Oy. Let the co-ordinates of P and Q 
be (#!3>i) (#2jy 2 ) respectively. We proceed to find an 
expression for the length PQ in terms of these co-ordi- 
nates. Draw PM, QM' perpendicular to Ox, and PN 
perpendicular to QM. Then, since PQN is a right- 
angled triangle, we have by the theorem of Pythagoras 

84 



THE FOUR-DIMENSIONAL CONTINUUM 



But 
and 



P() 2 = PN' 2 

PN = OM' - OM 



- - AS _ AS 



QN = QM' - NM' 
= QM' - PM 



therefore P<? 2 = (x 2 - xj* + (y 2 - yj*. 

Now whatever axes we use to locate P and Q, pro- 
vided we keep to the same plane, an expression of the 
form 




FIG. 25. 

will always have the same value, since it always repre- 
sents the distance between P and Q, and this distance 
does not depend upon the choice of axes. Take, for 
example, new axes 0%' , Oy' (Fig. 25), inclined to the 
original ones, which are shown in dotted lines. If 
( x i'yi) (xz'y*') De the new co-ordinates, and we make 
the same construction as before, represented by refer- 
ence letters m, m' and n, corresponding to M, M' and N, 
we shall have 

PQ* = p n * + Qn* 



86 THE THEORY OF RELATIVITY 

In the same way we may try any other set of rect- 
angular of axes in the same plane (we need not even 
keep to the same origin), and we shall find that the 
expression having the form 



always preserves the same invariable magnitude what- 
ever set of axes may be used. 

An expression of this kind is called an invariant. 
This does not necessarily mean the same thing as 
a constant, for the distance PQ may be anything 
we like ; but what we mean by invariant is that 
once having selected this distance, the expression 
(X 2 tfj) 2 + (^2 ~ JVi) 2 > which is always equal to it, is 
unaltered by any change of axes. 

In what follows we shall find occasion to apply this 
result to all kinds of distances such as PQ, whether 
these distances are measured along straight lines or 
curves. Now the above result is only strictly true if 
PQ is straight, but if we stipulate that PQ is always 
to be taken as very small, it will be substantially straight 
whether it forms part of a curve or not. A curve may, in 
fact, be regarded as the limiting case of a polygonal 
figure whose sides are infinitesimally short. It may be 
imagined as made up of a series of elementary straight 
parts placed end to end so that each element, as it is 
called, is inclined at an infinitesimal angle to the 
preceding one. 

With this understanding the following notation is 
adopted. It is usual to represent the length of any 
arc of a curve measured from some fixed point A by 
the letter s. Thus in Fig. 26Jwe might call^the arc 



THE FOUR-DIMENSIONAL CONTINUUM 87 



AP, $! and the arc AQ, s 2 . PQ is therefore s 2 s lf 
When PQ, according to our stipulation, is small enough 
to be regarded as straight, we agree to express this 
fact by calling it ds. Thus ds means an element of 
arc ; or, in fact, any elementary length. If, dropping 
the suffix, we call AP, s, then ds is the increment of s, 
or the elementary length which has to be added to AP 
to bring us to the adjacent point Q. The symbol 
simply means a small change in s. If (x, y) be the 
co-ordinates of P, the corresponding changes PN and 
QN in % and y are written dx and dy, conformably with 

'J 




FIG. 26. 

the notation ds for the length PQ. Thus if ds repre- 
sents an elementary length in any direction, and dx 
and dy represent the corresponding elementary lengths 
measured parallel to the axes of reference, we have 

ds 2 = dx 2 -f- dy* 
as the equivalent of 



when all the distances are small, and PQ can therefore 
be considered straight. 

It is to be noted that there is no special significance 
in the notation ds beyond the fact that ds means a very 



88 



THE THEORY OF RELATIVITY 



short straight length. The symbol, for example, does 
not mean d multiplied by s. The letter d may be read 
as the initial letter of the word " difference ". ds, dx, 
and dy may conveniently be called line elements, ds is 
the general expression for a line element in any direc- 
tion, dx and dy are line elements parallel to the axes. 
The relation 

ds 2 = dx* + dy 2 

means that dx* + dy* is the equivalent in any reference 
frame of the square of the line element in any direction. 
If we represent any change of axes by dashed letters 
x', y', we shall have always dx* + dy 2 = dx' 2 +.dy r *. 




FIG. 27. 

We note (i) that the expression dx* + dy 2 refers to a 
frame of reference in two dimensions of space only ; 
(2) that it consists of two terms, dx 2 and dy* ; (3) that it 
is invariant. 

Next consider the case of three dimensions of space. 

Take any reference frame consisting of rectangular 
axes Ox, Oy, Oz, and let PQ be a line drawn in any 
direction. Let the co-ordinates of P be (x 1 y 1 z 1 ) and 
those of Q(x 2 y z 2 ). Then, as the reader can satisfy 
himself,* 

* If the reader has any difficulty about this statement he is 
advised to make a paper model. His difficulties will then disappear. 



THE FOUR-DIMENSIONAL CONTINUUM 89 



or, with the same stipulation as before, 
ds 2 = dx* + dy* + d&. 

We note that dx 2 + dy z now represents the square of 
PQ' t the projection of PQ on the plane x Oy, and it 
therefore varies in general with the inclination of this 
plane with reference to PQ. It depends upon the 
planes of reference chosen, and is therefore not 
orientation of the invariant. The invariant in this 
case is the full expression for the square of the line 
element ds, or 

dx* + dy* + dz 2 , 

which consists of three terms corresponding to the three 
spatial dimensions. 

Comparing the cases of two and three dimensions of 
space we see that the invariant expression for the square 
of the general line element contains as many terms as the 
number of dimensions of space under consideration. 
This suggests that if any transformation of co-ordinates 
(or reference frame, or point of view, as the reader 
prefers) introduces additional quantities and corre- 
sponding additional terms into the expression, we know 
that we have introduced as many additional dimensions. 
We shall now apply this to the Lorentz transformation. 
Let us take the expression for the square of the distance 
PQ between two points 



and test it for invariance by applying the appropriate 
formulae of set (7), Chapter IX that is, we put 



90 THE THEORY OF RELATIVITY 



= Z-, 



and make similar substitutions for x z> y 2 , and z 2 . We 
then get 



This obviously depends upon u, which enters twice over, 
once as the multiplier of t z f ^' and again as a com- 
ponent of 0, or i/V 1 w 2 /c 2 . It is therefore a new form. 
The original form in the new co-ordinates namely, 

(*,' - V) 2 + (*' - 3V) 2 + (V - O 1 



does not represent the square of PQ, and is therefore 
not invariant. 

But suppose we test 



instead. We shall now have to use the relation 
t flUf H -- jM and a corresponding relation for t z . 
The expression becomes 



Remembering that /3 2 = i/(i w 2 /c 2 ), this becomes, 
after some reduction, 

(V - O 2 + OV -j'zT + (V - V) 1 - *(*,' - </). 



THE FOUR-DIMENSIONAL CONTINUUM 91 

which is of exactly the same form as that with which 
we started, and is independent of u. 

With the notation which has been used already, we 
say that dx 2 + dy 2 -f- dz 2 c 2 dt 2 is invariant. We may 
continue to call it ds, but ds no longer represents a line. 
We have, in fact, introduced time into the specification 
of P and Q. P and Q are therefore what we called 
events, or point-events, in Chapter III, and ds is not 
the distance between two points, but the interval 
between two events. 

We thus see that the use of the Lorentz transforma- 
tion involves the new term c 2 dt 2 in the expression 
for ds. This is analogous to the introduction of another 
dimension, with the difference that the new term is 
subtractive instead of additive. It might be thought 
that the factor c constituted another point of differ- 
ence, but since we may choose our units of length and 
time as we please, we may select them so that c, the 
velocity of light, is unity. With this understanding the 
invariant expression is 

dx 2 + dy 2 + dz 2 - dt 2 . 

If we represent as before any change of reference 
system using rectangular co-ordinates by dashed letters, 
we shall have always 

dx 2 + dy 2 + dz 2 - dt 2 = dx' 2 + dy'* + dz'* - dt' 2 . 

We have to remember that the use of the Lorentz 
transformation is not a matter of choice or convenience. 
It is a necessity if the statements of differently circum- 
stanced observers are to be correlated, and therefore 
this fourth dimension is forced upon the physicist. He 
has no option in the matter but to accept the fact that 



92 THE THEORY OF RELATIVITY 

he has to deal, not with space of three dimensions and 
an independent time, but with a state of affairs in 
which all four are inseparably associated. He is obliged 
to realize that lengths and times as manifested to him 
are not absolute properties of bodies existing indepen- 
dently of him, but relations between himself and some 
fundamental entity in which time plays the part of a 
dimension. We are thus brought back to the point at 
which we left the relativist in Chapter I, and we are 
in a position to see what he meant by saying that 
mathematical processes would distinguish time from 
length, breadth, and height. The distinction consists 
in the minus sign prefixed to the time symbol. We 
also see that the main feature of the statement of 
physical laws agreeably with the Restricted Principle 
of Relativity must be the use of a reference system in 
this unfamiliar compound of space and time. It is 
only by analogy that the word " space " can be applied 
to this concept. The concept was arrived at by an 
application of considerations suggested by the step from 
two dimensions of space to three, and there is therefore 
something to be said for retaining the word and extend- 
ing its meaning. But it is better to use some other 
word, retaining " space " for its ordinary usage. The 
word " continuum " suggests itself for reasons which 
will presently appear. 

All attempts to form a picture of a figure in a con- 
tinuum of four or more dimensions are in the writer's 
opinion futile. The mathematician is in no difficulty, 
for he is able to express by means of his formulae all 
properties relevant to his purposes without the neces- 
sity of forming a picture ; a picture would not help 



THE FOUR-DIMENSIONAL CONTINUUM 93 

him materially. But this resource is not open to those 
without mathematical training. Those properties of 
things which the mathematician can discard as irrele- 
vant are often the very ones upon which others rely 
for their concepts, and so the plain man is puzzled 
when he hears the mathematician talk of four dimen- 
sions. He does not realize that what the mathe- 
matician is thinking of is things which he can put down 
in a formula, while he himself is thinking of things out 
of which he can make a picture, and that these are not 
necessarily the same. It does not occur to him that 
in the matter of picture-making the mathematician 
may be in as great difficulties as himself. But though 
a picture may be just as impossible to the one as to the 
other, the mathematician has in his formulae perfectly 
adequate means of representing, though not of pictur- 
ing, all he wants. 

For example, we have seen that in two dimensional 
space the expression for ds 2 is dx 2 + dy 2 , and is 
invariant. This is an essential property of space of 
two dimensions, which may therefore be denned as 
that condition in which this expression is invariant. 
This is merely a formula, but it is all the mathematician 
requires. From the mere fact that dx 2 -f dy 2 is in- 
variant, the mathematician can derive the whole of the 
geometry of two dimensional space, and it is more or 
less incidental that in this particular case the expression 
can be interpreted as the square on a line. Similarly 
space of three dimensions can be denned as that con- 
dition in which the three-term expression dx 2 + dy 2 + dz 2 
is invariant, and again no picture is required. Proceed- 
ing in the same way we can say that a four-dimensional 



94 THE THEORY OF RELATIVITY 

continuum is a condition in which a four-term ex- 
pression of the same kind is invariant. Now this is 
perfectly intelligible as far as it goes, and it goes far 
enough to contain positively all that the mathematician 
wants. Similarly he might proceed to define an 
tt-dimensional continuum in a manner perfectly ade- 
quate for his own purposes. He might, of course, have 
a not unnatural curiosity to know what things would 
look like in such a continuum, but this is only a matter 
of mild interest. It is, on the other hand, everything 
to the plain man, to whom the formula is nothing. 

Events whose co-ordinates differ by very little from 
another are said to be adjacent, and it is clear that 
events may occur so closely in succession, and so near 
together in space, as to form a series as nearly con- 
tinuous as we please. Hence the name continuum. 
The physical history of any object is such a series of 
events. It is called a world-line. This term is a literal 
translation of the German " weltlinie ". When the 
world-lines of objects intersect those of observers the 
objects become manifest as phenomena. 
. 

Summary. The expression for the square of the line 
element ds in a rectangular reference frame consists of 
the sum of series of terms dx 2 -f dy* -f- . . . This ex- 
pression is invariant that is to say, it suffers no change 
in magnitude through change of axes of reference. 
There are as many terms in the expression as there are 
dimensions in the space under consideration, and there- 
fore, since the Lorentz transformation introduces four 
terms into it, a four-dimensional continuum is indicated 



THE FOUR-DIMENSIONAL CONTINUUM 95 

for the statement of laws conformably to the principle 
of relativity. Time is distinguished from the other 
dimensions of this continuum by the sign prefixed to 
the corresponding term. All geometry can be de- 
veloped from the bare fact of the invariance of the 
expression for the square of the line element in terms 
of the co-ordinates without help from diagrams, and 
the number of dimensions is no obstacle to mathe- 
matical representation. A convenient notation for 
expressing the line element is explained. Thus, for 
brevity, dx 2 + dy 2 + dz 2 is written instead of 

(* 2 -*i) 2 + b' 2 -^i) 2 + fe-*i) 2 
when the differences x 2 x lt etc., are small. 



CHAPTER XII 
THE GENERAL PRINCIPLE OF RELATIVITY 

WE now resume the main subject of the expres- 
sion of physical laws independently of particular 
observers where it was left off at the end of Chapter IX. 
We are about to start on an entirely new inquiry, 
which, though suggested by what has preceded, has 
no logical connexion with it. Excepting that the 
general attitude of mind and way of looking at things, 
more especially in respect of the four-dimensional con- 
tinuum, forms a good and practically an indispensable 
preparation for what follows, logically we might have 
commenced the whole subject with the general theory, 
which we are now about to consider. The general 
theory is not deduced from the restricted theory, 
though the restricted theory constitutes a particular, 
or limiting, case of the general theory, of which fact 
advantage is taken, as will appear in the sequel.* In 
developing the general theory we are not going to make 
use of any of the assumptions or postulates relating to 
the velocity of light, and the inability of an observer 
to detect his movement in the aether, nor any of the 
deductions therefrom. In fact, we shall see that the 

* Chapter XVIII. 
96 



THE GENERAL PRINCIPLE OF RELATIVITY 97 

velocity of light is constant only in the absence of a 
field of gravitational force. 

The restricted theory, however, suggests a more 
general theory in one way. No statement of physical 
law can be regarded as wholly satisfactory so long as 
it is confined to unaccelerated systems of reference. 
The phenomena themselves have no such preference. 
Physical agencies act whether the regions in which they 
reside are accelerated or not, and to confine the state- 
ments which represent them to unaccelerated systems 
is an arbitrary restriction which cannot be accepted if 
it is by any means avoidable. We have, however, seen 
in Chapter VI the obstacle which the forces and accelera- 
tions peculiar to each individual accelerated system 
present to the adoption of such systems of reference, 
and it is necessary to add that all systems other than 
Cartesian suffer from the same disability whether they 
are subject to a bodily acceleration or not. Polar 
co-ordinates, and Gaussian co-ordinates generally, 
which were explained in Chapter III, involve the use 
of curves in the frame of reference, in part or wholly, 
and these curves import into the mathematical state- 
ments expressions for what are termed centrifugal 
forces.* It is, unfortunately, not possible with the 
limited amount of mathematics at our disposal to 
illustrate this point, and therefore the general reader 
must be asked to accept the fact that no systems other 
than Galilean that is, unaccelerated Cartesian sys- 
tems can be used without importing into the formulae 

* The reader who understands Particle Dynamics will see the 
point at once. It is obvious from the expressions for accelerations 
in Polar systems. 
7 



98 THE THEORY OF RELATIVITY 

expressions for forces which are peculiar to each system, 
and which may therefore be expected to upset anything 
in the nature of general statements. 

The illustrations which we shall be compelled to use 
will be taken from cases in which the reference systems 
are subject to bodily acceleration. It will, however, 
be made clear that these cases are particular instances 
of change of co-ordinates, and the reader must therefore 
understand that forces are in general artificially induced 
by any change of co-ordinates. All reference systems 
in which curvilinear co-ordinates are used count as 
accelerated systems, though the propriety of so regard- 
ing them may not be as obvious as when bodily accelera- 
tion of the system takes place. 

There appears, therefore, to be an insuperable 
obstacle to the statement of physical laws in such a 
way as to be common to all observers whatever their 
circumstances, but it may be shown that the difficulty 
disappears if what is called the " Principle of Equival- 
ence " is granted. It is found that this principle 
enables us to act upon the postulate that All Gaussian 
systems are equally applicable for the statement of general 
physical laws. This postulate is the General Principle 
of Relativity. As Gaussian systems mean practically 
any reference systems whatever, statements of laws 
with reference to them will be of the most general 
character. We shall return to the application of this 
principle after the argument has been further de- 
veloped.* 

We have seen that in the general case, a change or 
transformation from one reference system to another 
* Chapter XX, 



THE GENERAL PRINCIPLE OF RELATIVITY 99 

involves the introduction of forces. The principle of 
equivalence may, with sufficient accuracy for present 
purposes, be stated as follows : A gravitational field of 
force is exactly equivalent to a field of force introduced by 
a transformation of the co-ordinates of reference, so that 
by no possible experiment can we distinguish between 
them* 

It has been shown that in order to carry into effect 
the mechanical principle of relativity it was necessary 
to assume that lengths and times were unaltered by 
relative motion, these suppositions being embodied in 
the scheme of transformation (i), Chapter VI ; also, 
that the restricted principle of relativity required the 
supposition that lengths and times altered in a special 
way with relative motion, which suppositions were 
embodied in the Lorentz transformation, and this in 
turn involved the reference of phenomena to a four- 
dimensional continuum. Now the principle of equival- 
ence stands in the same relation to the general principle 
of relativity as the suppositions respecting lengths and 
times stand to the mechanical and restricted principles. 
It is required in order to carry the general principle into 
effect. The principle of equivalence, together with the 
idea of the four-dimensional continuum, are the founda- 
tion of the general theory. The last chapter dealt with 
the four-dimensional continuum ; those immediately 
following will deal with the principle of equivalence. 

It will be found that the discussion of the principle 
of equivalence will disclose the remarkable fact that 
gravitational forces and the geometrical properties of 

* Eddington, "Report on the Relativity Theory of Gravitation," 
pp. 19, 43 ; and "Space, Time, and Gravitation," p. 76. 



ioo THE THEORY OF RELATIVITY 

the regions or fields in which these forces occur, are but 
different aspects of the same thing. This relationship, 
it will be shown, forms the basis of a new law of gravita- 
tion. It will thus be seen that gravitation possesses 
an importance hitherto unsuspected. Physical agen- 
cies, of whatever kind, necessarily conform to the 
geometry of the region in which they act, and if geo- 
metry and gravitation are merely different ways of 
viewing the same set of facts, it is clear that gravitation 
likewise controls these agencies. On account of the 
pre-eminent position assumed by gravitation, the 
General Theory of Relativity is also called the Gravita- 
tional Theory. The remainder of the book will deal 
with these matters. 



Summary. The restriction of statements of physical 
law to Galilean systems is arbitrary, but the introduc- 
tion of new forces is an obstacle to the use of other 
systems. Any Gaussian system, however, can be used, 
if advantage is taken of the principle of equivalence. 
This principle states that gravitational fields and fields 
of force artificially induced by change of co-ordinates 
are equivalent. In stating laws in conformity with the 
general principle, phenomena are regarded as occur- 
ring in a four-dimensional continuum. Geometry and 
gravitation are inter-related. 



CHAPTER XIII 
ROTATING SYSTEMS 

AS the first illustration of the Principle of Equival- 
ence we shall consider the forces induced by 
rotation and the corresponding geometrical relations. 

The parts of a rotating body are subject to accelera- 
tions in lines directed towards the centre or axis of 
rotation. These accelerations arise in accordance with 
Newton's first law of motion, which states that every 
body persists in its state of rest or of uniform motion 
in a straight line unless acted on by a force. The 
natural tendency of the body is to move along a tangent 
to the circle which it describes, and this tendency is 
what is called the inertia of the body. Newton's first 
law is the definition of inertia. 

Imagine an observer situated on a platform made to 
rotate in its own plane with constant angular velocity 
that is, equal angles are described in equal times. We 
shall suppose the platform to be rough, so that the 
observer can keep his footing, and we shall further 
suppose the platform to be located somewhere remote 
from all other objects, and to be so circumstanced 
otherwise that he has no direct means of perceiving the 
rotation. If the platform is in the earth's gravitational 

101 



102 THE THEORY OF RELATIVITY 

field we must suppose it to be horizontal, so that gravity 
has no moving effect on any of the objects on its 
surface, but it is better to think of the platform as 
situated somewhere away in space beyond any gravita- 
tional fields due to the presence of extraneous bodies. 
The observer will share the rotation of the platform. 
His inertia will assert itself, and he will therefore be 
subject to the acceleration to which, as we have seen, 
all bodies are subject under such conditions, but as he 
is unaware of the rotation he will not attribute his 
acceleration to this cause. What he will notice is that 
as he walks about on the platform he is continually 
urged away from one particular point, which point is, 
in fact, the centre of rotation, though he does not 
recognize it as such. He will also notice that this force 
acting upon him is exactly proportional to his distance 
from that point, at which point he finds that it vanishes. 
His experiments will also show him that the force has 
the same accelerating effect on all bodies alike. What- 
ever their mass or material, they will always gain the 
same outward velocity in the same time, provided only 
that they are at the same distance from the centre, and 
that he is careful to remove or allow for all agencies 
such as friction * or air resistance which might mask 
the effect of the force upon the body. In accordance 
with Newton's second law of motion, this equality of 

* As it may be asked how the platform could communicate to a 
body the rotation necessary to set up the centrifugal force if there 
is no friction, it is suggested that the motion might be communicated 
by a smooth guide directed towards the point of no force. This 
would supply the requisite constraint, while allowing the body to 
move radially. 



ROTATING SYSTEMS 103 

accelerating effect carries with it the fact that the 
force acts upon bodies with an intensity proportional 
to their masses.* If he supposes the seat of the force 
to be at the central point, and he attempts to screen 
off the force from the body by the interposition of 
other objects, he will find that the operation of the 
force is unaltered. In fact, this force presents the two 
essential features of gravitation : (i) it has the same 
accelerating effect upon all bodies whatever their mass 
or constitution, and acts upon them with an intensity 
proportional to the mass ; and (2) it cannot be screened 
off. It is true that it is directed away from a centre 
instead of towards one, unlike ordinary gravitation, 
and it acts according to a different law, but this does 
not affect the main features just mentioned. The 
observer, in fact, believes himself to be in a gravitational 
field. 

Let us now turn from the effects of forces on the 
platform to its geometrical relations. Suppose that an 
airplane flies over it, so that unknown to the observer 
the path is a straight line relatively to some other 
observer, who is using an unaccelerated reference frame. 
What will the path look like to the man on the plat- 
form ? This will, of course, depend to some extent on 
his position on the platform, but for simplicity we shall 
suppose him to stand at the centre. He can identify 
this point, not by reference to the rotation, for he is 
ignorant of it, but as the point at which he feels no 

* See Chapter IV. If m, m' and F, F' are respectively the masses 
of two bodies and the forces acting on them, and/ the common 
acceleration, we have by Newton's second law, F = mf, and F' = 
m'f; so that FjF ' = m[m'. 



104 



THE THEORY OF RELATIVITY 



force. The annexed figures show the paths relatively 
to the two observers. 

Fig. 28 shows the path AB plotted relatively to an 
observer 0, whose position coincides with the centre of 
rotation of the rotating platform, but who refers the 
movement of the airplane to an unaccelerated reference 
frame, Ox, Oy. This frame will have no rotation since 



y 





JC 



FIG. 28. 

it is unaccelerated, and we may consider it fixed. Let 
AB, which passes through a point directly over 0, be 
the path of the airplane, and let the numerals i, 2, etc., 
represent its successive positions at equal time intervals, 
say two seconds. It is required to find what appear- 
ance the path will present to the observer on the 
rotating platform, who will refer everything to a 
reference frame fixed on his platform, as he is unaware 



ROTATING SYSTEMS 



105 



of the rotation. Let this frame be OX, OY. Although 
0' actually coincides with 0, the reference frame of 0' 
is shown in a separate diagram, Fig. 29, for clearness. 
We will suppose the platform to rotate once in twelve 
seconds, or 30 in one second. Let the position i of 
the airplane correspond to the time when the two refer- 
ence frames coincide. We can plot the apparent path 




FIG. 29. 

on the reference frame OX, OY by pinning two pieces 
of paper on a drawing-board by a drawing-pin repre- 
senting the coincident points 0, 0' . The upper piece 
of paper carries a diagram of the reference frame Ox, Oy, 
and the lower one a diagram of OX, OY. We start 
with both reference frames coincident, and prick 
through the point i in the upper paper so as to make 



io6 THE THEORY OF RELATIVITY 

a corresponding mark in the lower. We then turn the 
lower paper through 60 relatively to the upper and 
prick through the point 2 in a similar way, and so on. 
We remove the upper paper and draw a curve through 
the marks on the lower, and the curve will represent 
the path of the airplane as it appears to 0'. 

We now provide the observer with a non-rotating 
platform on to which he can step. As he does so the 
whole aspect of affairs changes. The gravitational 
field disappears as far as he is concerned. The force 
which he formerly attributed to gravitation is now 
interpreted as the tendency of bodies to pursue straight 
paths that is, as inertia. At the same time, the 
complicated curved path of the airplane becomes a 
simple straight line described uniformly. He steps 
back again on to the moving platform and the former 
state of things is restored forthwith ; the gravitational 
field reappears, inertial masses become gravitational 
masses, and the geometry of the field alters corre- 
spondingly. 

From this we learn several important things : (i) All 
these changes are brought about simply by changing 
the point of view of the observer. While he is on the 
moving platform he is, in the language of the mathe- 
matician, referring all the circumstances to rotating 
axes ; while he is on the non-rotating platform he is 
referring them to fixed axes, and what he does when he 
steps from one to the other is essentially nothing more 
than to change over from one set of axes to another, 
or, as the mathematician would say, the changes are 
brought about by a transformation of co-ordinates. 
He has changed from an accelerated (in this instance, 



ROTATING SYSTEMS 107 

a rotating) system to an unaccelerated one, and the 
whole set of changes follow accordingly. (2) The 
physical quantity called mass is interpreted by him in 
the one case to be gravitational mass, and in the other 
inertial mass. The two are one and the same thing 
looked at in different ways. (3) Corresponding to the 
gravitational field are geometrical relations peculiar to 
itself. (4) In this particular instance the gravitational 
field can be completely extinguished by a change of 
axes. A suitable mathematical transformation trans- 
forms the gravitational field out of existence. (5) The 
gravitational field arises because bodies are guided in 
a particular way. Bodies rotating with the platform 
are guided so that they describe circles, with the result 
that a force arises which has all the essential properties 
of a gravitational force. While the man is on the 
rotating disk he looks on the body as acted on by a 
force at a distance ; he looks on a certain point on 
the disk as a centre of repulsion, as we regard the 
centre of the earth or of the sun as a centre of attraction. 
His attention is concentrated on this centre and not 
on the course of a body. When he is off the disk on 
the other hand, his attention is concentrated on the 
circular course, and he ascribes the force, previously 
thought of as due to some distant agency, to the fact 
that the body is constrained to move in that particular 
way. 

Summary. Forces arising on a rotating system from 
centrifugal action are indistinguishable in principle 
from gravitational forces. The geometrical relations of 
the system show a correspondence with the forces. 



io8 THE THEORY OF RELATIVITY 

Mass can be interpreted either as gravitational or 
inertial, according to the point of view. Whether the 
mechanical conditions on such a system are to be 
regarded as gravitational or due to centrifugal action 
is purely a matter of choice of co-ordinates. A gravi- 
tational field artificially induced by rotation can be 
transformed away completely. 



CHAPTER XIV 
TRANSLATION 

last chapter dealt with a case of an accelerated 
JL system in which the bodies composing it were 
subject to an acceleration transverse to their line of 
motion. The present one will deal with a system 
subject to an acceleration in the line of motion. The 
word " acceleration " will therefore approximate in 
meaning to its popular sense of increase of speed, 
though it will include decrease. The extension of the 
use of the word to transverse effects is a refinement 
which has nothing corresponding to it in popular usage. 
As has been shown, this extension is justifiable, since 
the same agency is at work in both cases. All forces 
produce acceleration in their line of action only, but if 
a body has already a velocity in some other direction 
the speed-increasing effect of the force is more or less 
masked, and may be completely masked, so that 
acceleration may, as we have already seen, show itself 
in a bending of the path of the body only. 

The subject matter of the present chapter is dealt 
with in Chapter XX of Einstein's book in a way which 
leaves no room for improvement in clearness or sim- 
plicity. What we have to say, therefore, can only be 
for the most part paraphrase. 

109 



no THE THEORY OF RELATIVITY 

Let us imagine ourselves in a closed box like a room 
situated in some region of space where there is no 
gravitational field. There will be no such thing as 
weight. If we put an object in mid -air and leave it, 
it will remain where we put it, but the slightest touch 
will send it moving off uniformly in a straight line in 
the direction in which it was pushed, with greater or 
less speed, according to the strength of the touch. 
When the object encounters a boundary of the room 
(we cannot now say wall, or floor, or ceiling, for there 
is no up or down or sideways) it stays there, if it 
is not elastic. If it is perfectly elastic it will rebound 
with the same velocity and continue in perpetual move- 
ment. When we push the object we ourselves will 
recoil and continue to recoil, until we are brought up 
by a boundary. Indeed, the least push against a side 
of the box or against any object on our part will set 
us going perhaps through the air so that if we want 
to stay in any particular place we must tie ourselves 
there. 

But now suppose that a rope is hooked on to the box 
outside, and some being, no matter how, pulls on the 
rope so as to give the box a uniform acceleration. 
Immediately everything in the box which is not already 
at the side remote from the rope attachment gets left 
behind as the box moves forward, and from the inside 
things have the appearance of falling towards this side 
now the floor with uniform acceleration. All objects 
within the box which are not already on the " floor " 
are affected alike. If there is no air in the box they 
all fall with the same acceleration, just as in a gravita- 
tional field. If there is air it is carried along with the 



TRANSLATION in 

box, and partly carries the objects with it to a greater 
or less degree, according to their density, size, and 
shape, thus exactly imitating the effect of air resistance 
on falling bodies. Anyone standing on the floor when 
the motion started will immediately feel the sensation 
of weight, and will have to support himself by his legs. 
If he tries to prevent things from reaching the floor 
falling, as he thinks he will find that the things which 
are hardest to support are those which were hardest to 
move before the box began to accelerate. If he is at 
a loss to account for these phenomena, and he chances 
to look towards what is now the ceiling, he may see 
part of the hook attachment, and he may very well 
conclude that he is suspended by it in a gravitational 
field ; it is quite likely that it may never strike him 
that the whole thing is simply the result of the box 
having been set in motion with an ever-quickening 
velocity. Objects which he thinks are falling are simply 
being left behind as the box moves. Again, if he 
attaches a body to the ceiling by a string, the string 
will be put in tension, due, as the observer inside 
thinks, to the weight of the body, but, as the being 
outside thinks, to the fact that the box is pulling the 
body along with it. 

Accompanying all this there is a corresponding dis- 
tortion of the paths of moving objects. Bodies thrown 
across the box which formerly described straight lines, 
now describe parabolas, or, if air resistance is taken 
into account, trajectories exactly like those in a gravita- 
tional field. 

In short, we get a set of circumstances exactly 
parallel to those examined in the last chapter. There 



ii2 THE THEORY OF RELATIVITY 

is no distinguishable difference between the phenomena 
inside the box whether they are regarded as due to a 
gravitational field or to an acceleration impressed on 
the box. To a person inside they have the appearance 
of occurring in a gravitational field ; to a person out- 
side they are due to inertia. To both, inertial and 
gravitational mass are the same. Which way we regard 
the phenomena is indifferent ; it is all a question of 
point of view, or, to put it mathematically, of choice 
of axes. The observer inside the box refers phenomena 
to a reference frame moving with the box, and fixed 
relatively to himself ; the observer outside refers them 
to a frame fixed with reference to himself. 



Summary. Systems in which acceleration takes the 
form of change of speed only, exhibit the same features 
as those described in the last chapter with reference to 
rotating systems. The phenomena can be interpreted 
as due either to acceleration or to an artificial gravita- 
tional field. The gravitational field can be wholly 
transformed away by a change of axes, and a change 
of axes from one point of view to the other is accom- 
panied by a parallel change in geometry. 



CHAPTER XV 
NATURAL GRAVITATIONAL FIELDS 

A CHARACTERISTIC feature of the instances con- 
-iVsidered in the last two chapters is that the gra- 
vitational field can be completely destroyed, as such, 
by a suitable transformation to another frame of 
reference. All forces attributed to gravitation before 
the transformation are attributed to inertia after it, 
and this effect extends to the whole of the field. We 
have now to inquire whether gravitational fields such 
as occur in nature the earth's gravitational field, for 
example can be transformed out of existence by a 
similar process. 

Imagine an observer enclosed in a box as before, but 
falling freely in the earth's field. An observer on earth, 
who regards the occurrence from the point of view of a 
fixed earth that is, he refers it to a fixed frame of 
reference is conscious of the earth's gravitational 
force, and he attributes the motion of the box to that 
cause. But an observer inside the box is not conscious 
of any force whatever. The acceleration acts equally 
on the box itself and all the things inside it, including 
the observer himself. A body placed in mid-air in the 
box will remain in the same position relatively to the 
box, and the observer will feel no weight. If he wishes 
8 113 



THE THEORY OF RELATIVITY 



to remain in one position he has to fasten himself 
there ; and a body thrown from one side to the other 
will describe apparently a straight line. As far as the 
interior of the box is concerned, matters are exactly as 
described in the last chapter before the box, which was 
then considered, had been given an acceleration. 

But now suppose the size of the box to increase, as 
shown in Fig. 30, so that the slope of radii drawn to 




FIG. 30. 

the centre of the earth from places near the sides of the 
box becomes appreciable. Objects at places such as 
A or B would actually fall, from the point of view of an 
observer on earth, along radii AO, or BO, but from the 
point of view of an observer situated at C inside the 
box, they would appear to move with an acceleration 
directed inwards towards him with a slightly upward 
tendency, as shown by the short arrows, while an object 
at C, where the observer is situated, would remain 



NATURAL GRAVITATIONAL FIELDS 115 

apparently in mid-air. If the box were large enough 
to include the whole earth so that objects at D, E, and 
F could be observed, it would be seen that the inward 
tendency increased as far as D and E and then 
diminished, while the upward tendency that is, the 
tendency towards the observer at C increased all the 
way round to F, where it would be exactly double of 
the actual downward tendency of the observer towards 
the earth.* The arrows show the approximate rela- 
tive magnitudes and directions of these tendencies as 
they appear to the observer in the box. 

It seems then that though it is possible by using a 
suitable system of reference to transform away gravita- 
tion in a small region which may be anywhere, since 
we have not specified any particular position for the 
observer this transformation, so far from annihilating 
the whole field, only aggravates the effects of gravitation 
in the remainder. Gravitation can, so to speak, be 
smoothed out in one place only to appear with greater 
intensity in another. f But still we have the fact that 
anyone can, in his own neighbourhood, produce all the 
effects considered in the last two chapters. By merely 
altering his point of view an observer can, in any small 
region, regard a force as either inertial or gravitational, 
and, as before, his geometry will follow his choice. 

In the last two chapters no stipulation was made as 

* Readers familiar with the parallelogram of accelerations can 
easily verify these statements. 

t Newton uses exactly this transformation in that famous proposi- 
tion LXVI, Book I, of the "Principia". See also Herschel's 
"Outlines of Astronomy," 1878 Edition, 610, p. 415; also 
Proctor, "Old and New Astronomy," pp. 207, 208. 



n6 THE THEORY OF RELATIVITY 

to the size either of the revolving platform or of the 
observer's box ; the transformations there considered 
included all space. We therefore have to distinguish 
between the cases in which gravitation can be annihi- 
lated everywhere, and those in which it can be 
annihilated throughout a small region only at a time. 
In the former case the gravitation is wholly attributable 
to choice of axes of reference ; in the latter it is due 
to the presence of attracting matter. We may dis- 
tinguish the two by calling the former an artificial 
gravitational field and the latter a natural one. 

These facts lead to the following re-statement of the 
principle of equivalence, which includes both cases : 

A gravitational field of force is precisely equivalent to 
an artificial one, so that in any small region it is impos- 
sible by any conceivable experiment to distinguish between 
them.* 

The limitation to small regions does not exclude cases 
where the whole gravitational field is artificial and can 
be extinguished by one transformation. For if all small 
regions happened to be alike, a transformation applied 
to one would affect all equally. 

We have seen that the fields of force which have 
been considered carry with them their own peculiar 
geometry. The principle of equivalence, therefore, in- 
volves a relation between gravitation and geometry and 
suggests the general possibility of a relation between 
the gravitational forces in any region and the geometry 
of that region, so that the specification of the one 
carries with it the specification of the other. Now a 
relation of this kind can be nothing else than a state- 
*Eddington, "Space, Time, and Gravitation," p. 76. 



NATURAL GRAVITATIONAL FIELDS 117 

ment of gravitational forces in terms of geometry ; it 
is, in fact, a Law of Gravitation, and it is to the possi- 
bility of obtaining such a law that the principle of 
equivalence directly points. 

An important question, however, arises. It is very 
well to say that if we can specify either the geometry 
or the gravitational forces we can specify the other, 
but how can we specify either when they are both 
unknown ? And what in particular is meant by 
specifying geometry ? In the next chapter a method 
of specifying geometry will be examined which will 
answer this latter question and allow us to proceed 
with the answer to the former. It will afterwards be 
seen that this method directly suggests an hypothesis 
connecting gravitation and geometry, which hypothesis 
is Einstein's Law of Gravitation. 

There is one possible difficulty with which it is 
desirable to deal before proceeding further. It may 
occur to the reader to ask what is the use of knowing 
either the geometry or the forces if our knowledge 
applies to a small region of the field only. It may not 
be clear to him how a transformation applied to one 
part only of the field can affect the whole. The answer 
to this question is that the properties, metrical and 
otherwise, of any region are taken as continuous that 
is, there are no sudden or capricious jumps between the 
properties of any particular region and the next. The 
geometrical properties which it is our object to specify 
will include the rate of variation of the various quan- 
tities from place to place, and therefore if we know the 
properties in one place we have the means, theoretically 
at least, of knowing them everywhere, much in the 



n8 THE THEORY OF RELATIVITY 

same way as a man who knows how much property he 
has at the present, and how fast it is increasing or 
dwindling, has the means of forecasting the probable 
state of his affairs in the future. If, therefore, there 
is any place where we cannot specify either the gravita- 
tional forces, or their rates of change, we cannot success- 
fully apply at that place any transformation with the 
object of destroying the gravitational field. Such a 
condition arises where attracting matter exists which 
is creating the gravitational field. We therefore exclude 
such places from the operation of the principle of equiva- 
lence. It will be seen that this point is not without 
importance in the sequel. 



Summary. A natural gravitational field can be extin- 
guished locally by a suitable change of point of view, 
but this local extinction modifies gravitational effects 
elsewhere. The principle of equivalence can be stated 
in terms appropriate to a natural gravitational field, 
and it indicates a relation between geometry and 
gravitation. A method of specifying geometry is 
required. A natural gravitational field cannot be 
transformed away where attracting matter exists. 



CHAPTER XVI 

GEOMETRY OF THE GRAVITATION THEORY 

IN order to describe the metrical properties of any 
region mathematicians have resorted to several 
methods, of which the system of Euclid is an example. 
The method, however, which now concerns us is based 
on the forms which the expression for the line element 
ds assumes under different conditions. In this chapter 
we shall confine our attention to space of two dimen- 
sions. The sense in which the word " dimensions " is 
used must not be confused with its popular meaning of 
magnitude. In the present chapter, and for the most 
part elsewhere in this book, the word has reference to 
the number of independent quantities which are re- 
quired to define a point or point-event. Thus, in the 
space of experience, three independent co-ordinates 
are required, and so we call it three-dimensional space. 
When we speak of the three dimensions of space we 
are simply referring in general terms to the measure- 
ments in three different directions necessary to locate 
a position. On a surface, plane or otherwise, two 
co-ordinates only are required when the surface has 
been decided upon, and we express this fact by saying 
that we are in, or are considering, two-dimensional 
space. The idea of curved two-dimensional space, how- 
ever, presents great difficulty to many who readily 
accept the term two-dimensional space as an intelligible 

119 



120 THE THEORY OF RELATIVITY 

description of a plane. They can understand the sup- 
pression of the third dimension in " flat-land/' but not, 
for example, on the surface of a sphere. The curvature 
of the sphere seems to obtrude itself, and force the third 
dimension on the attention. As a matter of fact, three 
dimensions are just as necessary to the appreciation of 
flatness as to that of curvature. Flatness is unmeaning 
without the corresponding idea of something which is 
not flat, and in the absence of the faculty of appreciating 
a third dimension, no comparison can be made. A 
" flat-land " being who has no such faculty, transferred 
from a plane to the surface of a sphere, would have no 
direct means of perceiving any change, though he 
might, theoretically at least, employ the indirect means 
described at the end of the present chapter. These 
means, however, do not help him to visualize a third 
dimension, nor do they alter the fact that positions in 
his space require two co-ordinates only to locate them. 
It was seen in Chapter XI that when plane rect- 
angular reference systems are used any short line ds 
in two dimensions can be expressed by the relation 
ds 2 = dx 2 -j- dy 2 .* This relation, however, holds good 
only for rectangular systems. If other systems are 
used, such as polar or Gaussian co-ordinates, the expres- 
sion for the line element becomes more elaborate. It 
is beyond the scope of this book to give this expression 

* This relation also holds good for cones and cylinders, which can 
be formed by curling a plane, e.g., rolling up a flat piece of paper. 
This refinement, however, is not required for present purposes. 
When curved surfaces are mentioned in the text it will be understood 
that reference is made to surfaces, such as that of a sphere, which 
cannot be flattened. 



GEOMETRY OF THE GRAVITATION THEORY 121 

in its most general form,* but it may be considered to 
be sufficiently exemplified for two dimensions by the 
relation 

ds* = gjxf + gjxf . . (8) 

which is a particular case of it. It will be observed 
that the Cartesian relation ds 2 dx 2 + dy 2 is the par- 
ticular form which this standard relation assumes when 
g 1 and g z are each equal to unity, and x and x 2 are 
identified respectively with % and y. The present 
chapter will consist of an examination of the meaning 
of the multipliers g t and g 2 when x l and x z signify other 
kinds of co-ordinates, such as the radius vector and 
vectorial angle f of polar co-ordinates, or latitude and 
longitude on a sphere. The immediate point which the 
following examples illustrate is a very simple one, 
namely, that the values of the g's in the relation (8) 
depend upon the system of reference used, and upon 
the curvature of the surface upon which the line element 
is drawn. This might, indeed, be taken as obvious, 
but the illustrations lead to some further considerations 
which are necessary to the development of the subject. 

PLANE POLAR CO-ORDINATES J 

Let the point P (Fig. 31) be located by the polar 
co-ordinates (r, 0} as explained in Chapter III, and let 
PQ be a line element extending from P to a point Q, 
whose co-ordinates are determined by adding on any 

* Eddington, " Space, Time, and Gravitation," p. 82. 

t See Chapter III for definitions of these terms. 

+ The reader will remember, from Chapter III, that these are 
nothing more than range and bearing, the radius vector, r, being 
the range, and the vectorial angle, 0, the bearing. 



122 THE THEORY OF RELATIVITY 

small quantity dr to r and a corresponding small 
quantity dO to 6, as shown in the figure. As we are 
considering small quantities only, it does not matter 
whether the element PQ forms part of a curve or not ; 
to all intents and purposes it is straight. Draw PR 
perpendicular to OQ. Now the smaller we take dO to 




FIG. 31. 

be, the closer together are P and R, so that as dO is very 
small, we may consider OP, or r, to be equal to OR. 
Thus OR = r, and therefore RQ = dr. For the same 
reason we may suppose PR to be equal to a small 
circular arc struck with as centre and OP as radius, 
and therefore the angle ROP or dO is RP/r radians.* 

* The reader who is unacquainted with the " circular measure " 
of angles may be informed that, 
according to this system, the unit 
of angular measurement is the 
angle POQ (Fig. 32), subtended 
at the centre, O y of a circle by an 
arc, PQ, equal in length to the 
radius, OQ. This angle is called 
a "radian". It is about 57 
degrees. Any angle is measured 
by its ratio to this unit, and is 
therefore equal to the ratio of the FIG. 32. 




GEOMETRY OF THE GRAVITATION THEORY 123 

Thus 

RP/r = M, or RP = rdO. 

But, since PQR is a right-angled triangle, we have 

PQ* = QR* + RP*, 
or ds 2 = dr* + r*d#*. 

Thus, if we identify r with x lt and (9 with # 2 in the 
standard relation, we see that g l = I, and g 2 = r 2 . 

We now notice a point which is most important for 
our purposes, and upon which due stress will be laid 
in the sequel. If we had chosen to refer the positions 
of P and Q to rectangular co-ordinates, we could, as 
has been seen, have expressed ds* in the form dxf -f- dxf 
by identifying x^ and x z with the Cartesian co-ordinates 
(x, y) of P. The transformation or change of point 
of view, as we have expressed it from a polar frame 
of reference to a Cartesian frame has the effect of 
replacing r 2 by unity. This is a general characteristic 
of plane geometry. In plane geometry every system 
of reference has its own peculiar corresponding expres- 
sion for the line element, but it is always possible by 
changing over to a rectangular Cartesian system to 
transform this expression for the line element into the 
form dxf -f dxf, thus reducing both & and g 2 to unity.* 

arc which subtends it to the radius. Thus, if ROQ or 6 be 
such an angle and a a radian, we have from the annexed figure, 

= ^ = fi * But PQ = OQt and a = J> by definition ' 

T>(~) 

and therefore Q ~. -~ 

* We omit to refer explicitly in the text to the more complicated 
cases where the product dxdy occurs in the line element. It 
disappears on transformation to a rectangular system. 



124 THE THEORY OF RELATIVITY 

LONGITUDE AND LATITUDE 

Consider the small triangle QPR drawn on a sphere, 
one-eighth part of which is shown in Fig. 32. The 
sides of the triangle, of which PQ, or ds, is the hypo- 
tenuse, are now necessarily curved, but this need not 




FIG. 33- 

trouble us, for, as before, the triangle is taken so small 
that the sides are substantially straight. We may take 
the earth as a specimen of a sphere, and speak through- 
out in geographical terms. Let AB be the equator and 
C the North Pole. AC is the zero meridian of longitude 
let us say the meridian of Greenwich. We will take 



GEOMETRY OF THE GRAVITATION THEORY 125 

P in north latitude \ and east longitude /. Thus, if 
we draw the meridian CPp through P, and join 0, the 
centre of the earth, to P and p, we have AOp = I, 
and POp = X. Take some point Q near P not on the 
same parallel of latitude, or meridian of longitude, 
draw the parallel of latitude DPR through P, and the 
meridian of longitude CQq through Q, and let them 
meet in R. Let the centre of the parallel of latitude 
DPR be at E, which will be on the earth's polar axis 
OG. Let dl and d\ be respectively the difference of 
longitude and latitude of P and Q, so that if we draw 
the remaining lines shown in the figure we have 

LAOp = /, LAOq = I + dl, LpOq = LPER = dl ; 

LPOp = LROq = X, LQOq = \ + d\, LQOR = d\. 
Let PE = p, and let OR, the radius of the earth, be a. 
Then by the rule of circular measure already explained, 



Thus PR = pdl. 

Also, LQOR (or d\) = ^ = 

Thus <?# = ad\. 

Then, since LQRP is a right angle, 

P 2 = PR 2 + () 2 , 
or ds* = p*dl* + a <fo 2 , 

if we call PQ t ds, according to previous practice, 
Whence, if we identify / with x and X with x 2 in the 
relation (8) we see that g 1 ~p 2 , and g 2 a 2 . 

The length p depends upon the radius of the sphere, 
and also upon the latitude X*. It obviously depends 

* Readers acquainted with trigonometry will see that p = a cos X. 



126 THE THEORY OF RELATIVITY 

upon the size of the sphere that is, upon the radius 
and therefore upon the curvature. It also depends 
upon the co-ordinate X, since if P were at the pole its 
magnitude would be zero, and if P were on the equator 
it would be equal to a. Thus, taken together, the 
multipliers g depend upon the system of reference 
chosen and also upon the curvature of the sphere. 

The same argument might be applied to line elements 
drawn on other surfaces, say the surface of an egg or 
a rugby football. We should in such cases get still 
more complicated expressions for the line element, but 
they would all illustrate the general fact that the g's 
in the standard expression for the line element depend 
upon the co-ordinates chosen and upon the curvature of 
the surface upon which the element is drawn. The 
same is true with reference to the plane, but in the case 
of the plane the entry of the curvature is not so obvious, 
since it is zero. 

It was noticed that in the geometry of the plane the 
standard expression for the line element could always 
be reduced to the form dx^ + dx 2 * by a suitable trans- 
formation, the g's becoming equal to unity. This is 
not the case with the sphere or with any other curved 
surface.* The impossibility of reducing the g's to unity 
is the necessary consequence of the impossibility of 
applying a curved surface to a plane so that they shall 
fit together without distortion. This transformation 
can, however, be effected at any one spot at a time for 
a small region on the surface. Suppose, for instance, 
that AB (Fig. 34) is the trace on the paper of a plane 

* Excepting cones and cylinders. See previous note. 



GEOMETRY OF THE GRAVITATION THEORY 127 

which touches any curved surface at P. It is clear 
that a small figure, say a triangle, drawn on the surface 
at P could be projected on to the tangent plane without 
substantial distortion. To all intents and purposes it 
might as well be drawn on the tangent plane. This 
being so, plane geometry can be applied to it, and the 
g's can be transformed away. But a figure drawn at 
Q some distance away from P could not be projected 
on to the tangent plane at P without considerable 
distortion. A transformation which would reduce the 
g's to unity might of course be applied at Q, but this 
would only produce distortion at P. 




FIG. 34. 

There is one important case in which this local trans- 
formation cannot be applied. It is necessary for its 
success that the curvature at the place of application 
should be continuous. If there is any sudden change, 
such as would be produced by a sharp ridge or elevation, 
the transformation could not be effected. Consider, 
for example, Fig. 35. If two parts PQ, PR of the 
surface meet at P, making a finite angle TPT', a figure 
near P on the part PQ would project without distortion 
on to the tangent plane PT, but a corresponding figure 
on the part PR would not. So also a figure near P on 
the part PR would project on to PT' without distortion* 
but not on to PT. If we take any other plane PS 



128 THE THEORY OF RELATIVITY 

through P as the plane of projection, figures on neither 
part could be projected without distortion. It is, there- 
fore, impossible to choose any frame of reference at 
such points as P in the neighbourhood of which the 
quantities g may be reduced to unity. It may be done, 
of course, for other points, but we are no longer able 
to say that it can be done at any point. This is only 
possible if the surface is continuously curved through- 
outthat is to say, no ridges, such as that at P, may 
occur anywhere. 




It is not necessary to the truth of the relation (8) in 
any of its forms that the line elements, or the small 
triangles related thereto, should be drawn on actual 
physically existing surfaces. The relation 

<& = 



where & and g 2 are given quantities involving x l and x 2 , 
is a condition which could be complied with if the 
surface on which the elementary figure is drawn were 
removed, leaving the figure itself in the air, but in the 
same position relative to the reference frame. The 
relation defines the kind of surface on to which a line 



GEOMETRY OF THE GRAVITATION THEORY 129 

element would fit if the surface actually existed. 
Looked at in this way the relation defines, not so much 
a surface, as the metrical properties of the region of 
space in which the relation obtains. It imparts a 
structure or curvature, as it were, to space which limits 
complete freedom of movement, and it has been found 
that all the g's appropriate to a particular conformation 
of space satisfy a corresponding set of conditions, no 
matter what admissible reference frame may be used.* 
Thus all the g's corresponding to plane geometry comply 
with the same set of conditions, whether the reference 
frame be rectangular, polar, or any other which can be 
used on a plane surface, those corresponding to spherical 
geometry comply with another set, and so on. The 
g's thus furnish a basis on which the geometry of any 
region can be worked out, that is to say, they specify 
the geometry. A being inhabiting two-dimensional 
space, and incapable of perceiving a third dimension, 
could nevertheless determine the curvature of his space 
though he cannot visualize the curvature. For he 
could measure up a number of triangles, and by com- 
paring the results determine empirically, in theory at 
least, the g's in the relation 



Summary. The number of dimensions of space is the 
number of co-ordinates necessary to determine a point 
in it. The geometrical properties of a region are 
specified by the form assumed by the expression for 

* Eddington ? " Space, Time, and Gravitation," Chapter V. 
9 



1 30 THE THEORY OF RELATIVITY 

the line element. For two dimensions of space the 
typical form is taken as g^dx-f -j- g 2 dx 2 z . It is shown by 
examples that the g's depend upon the reference frame 
chosen and upon the curvature of the surface on which 
the line element is situated. If it is on a plane the g's 
may be reduced to unity by a suitable transformation, 
but not if it is on a curved surface. This reduction 
may, however, be effected locally on any surface ex- 
cepting where the curvature is discontinuous. Local 
transformation produces distortion elsewhere. The 
expression for the line element determines the metri- 
cal properties of the region in which the element occurs, 
though no surface on which it might be fitted actually 
exists physically, and the geometry of the region is 
thus seen to be specified by the g's. Theoretically, the 
curvature of a space could be determined empirically 
without the need for visualizing it. 



CHAPTER XVII 

GEOMETRY OF THE GRAVITATION THEORY 

(continued) 

r I ^HE results of the last chapter for space of two 
-1 dimensions have their counterpart in space of 
three or more, but since this extension carries us beyond 
our visualizing powers it is necessary to inquire what 
means exist for representing these results, and in what 
sense the same terminology can be used. 

It was seen in the last chapter that in two-dimen- 
sional space a point is located by two independent quan- 
tities only, namely, its co-ordinates. This kind of space 
is therefore necessarily a surface, for we can proceed 
along a surface in any two independent directions we 
please, but we should have to leave the surface in order 
to proceed in a third independent direction. Every 
surface, whether plane or curved, therefore constitutes 
a two-dimensional region, and we may name it accord- 
ingly. Thus planes, or spheres, or other curved sur- 
faces are respectively plane, spherical, or curved two- 
dimensional regions. If the standard relation is such 
that figures can only be drawn which would fit on a 
curved surface, we say that we are in two-dimensional 
space curved in three dimensions. It is usual to call 
plane space Euclidian space, because, as has been seen, 
the theorem of Pythagoras, which is perhaps the most 
important and characteristic of Euclid's propositions, 



*32 THE THEORY OF RELATIVITY 

holds good with reference to it. This theorem does not 
hold good on a sphere or other curved surface, and 
curved spaces are therefore classed as non-Euclidian. 

In the last chapter the line element in two dimensions 
was expressed by the standard relation 

ds* = gidxf + g<4x z *, 

and the matter we have now to examine is the meaning 
to be attached to a corresponding relation if we proceed 
to three dimensions, thereby adding another term, 
which converts the relation into 

ds* = gjxf + gxf + g z dx*. 

K gi 2* an d 3 are all unity, or if they can be 
made unity by any change of reference system, the 
relation becomes 

ds* = dxj + dx 2 * + dx,*, 

which is the ordinary three-dimensional form of the 
theorem of Pythagoras. No question of curvature 
arises any more than in the corresponding two-dimen- 
sional case. We may therefore carry forward the same 
terminology as before and say that we are in Euclidian 
space of three-dimensions, and we can visualize the 
whole circumstances. But if the g's cannot be made 
independent of the co-ordinates, we have a situation 
parallel to the two-dimensional case, in which we had 
to resort to a curved surface in three-dimensional space 
on which to draw our two-dimensional figures two- 
dimensional, as has been seen, because defined with 
reference to two co-ordinates only. But in the present 
case we are in three dimensions already, and it would 
therefore seem necessary to describe the property 
imported into the expression by the extra term as 



GEOMETRY OF THE GRAVITATION THEORY 133 

curvature in space of four dimensions. This is ob- 
viously beyond our powers of visualizing, and we have 
to seek for a representation by a method corresponding 
to that which was applied in Chapter XI. We there- 
fore recur to the case of two dimensions, and ask what 
picture was presented to us when a plane was, as it 
were, distorted into a surface by the introduction of 
appropriate values of the g's into the expression for 
the line element, and what is the mathematical ex- 
pression of this picture. 

It was seen in the last chapter, or, rather, it was 
asserted for the mathematical proof is beyond the 
scope of this book that an expression could be found 
for the curvature in terms of the g's and any reference 
system which could be used.* We are thus presented 
with alternatives. We may either picture the curva- 
ture of a surface in three dimensions as its defect from 
flatness, or we may- define it by the mathematical 
formula which represents this defect. We cannot 
generalize the former, for pictures cannot be made in 
four dimensions, but there is no difficulty at all about 
generalizing the latter, beyond some additional com- 
plication. We therefore define the curvature of a 
surface by its mathematical expression, and generalize 
this expression according to the number of independent 
co-ordinates that is to say, dimensions which are 
being used. Thus if the g's in the relation 



cannot all be made independent of the co-ordinates, we 

* Mathematical readers will of course recognize that allusion is 
here made to Gauss' expression for curvature. 



134 THE THEORY OF RELATIVITY 

say that we are in three-dimensional space curved in 
four dimensions, and by curvature this generalized 
expression is meant. 

We may proceed in the same way for four or any 
number of dimensions. The present subject is limited 
to four. The standard relation may be written 

ds* - gjdxf + g 2 dx 2 * -f g s dx 3 * + &dx* * . (9) 
and we say that if any transformation will reduce all 
the g's to unity we are in Euclidian space of four 
dimensions ; or if any of them cannot be expressed 
except as depending on the co-ordinates, then we are 
in space of four dimensions curved in a fifth. 

Mathematically expressible conditions exist for the 
properties of continuity and discontinuity of curvature, 
which were considered in the last chapter. These 
expressions are perfectly adequate representations of 
the properties, though, of course, they are in no sense 
pictures. With this understanding such statements as 
that four-dimensional curved space can be reduced to 
four-dimensional Euclidian space, excepting at points 
of discontinuity, is perfectly intelligible when stated 
mathematically. It means simply that when the ex- 
pression for the curvature complies with the condition 
of continuity throughout any small region, the g's can 
be reduced to unity. As an illustration of the procedure 
we conclude the chapter with an explanation of the 
generalized meaning of the term " small region " which 
has been used above. 

* The reader may perhaps be reminded that this is not the most 
general expression for ds\ Ten terms are actually required, the 
remaining six containing products such as dx^dx^. The expression 
in the text is taken as a standard for illustrative purposes only. 



GEOMETRY OF THE GRAVITATION THEORY 135 

To arrive at this meaning we ask what is the charac- 
teristic mathematical feature of a small region of two 
or three dimensions. Simply this, that the co-ordinates 
of all points within it differ from one another by very 
little. A small region in any number of dimensions is, 
therefore, an aggregate of points whose co-ordinates 
differ by very little from one another. But a " point " 
in space of more than three dimensions has not been 
denned. In order to do this we notice that a point in 
two or three dimensions is determined by its co-ordi- 
nates (x lf x z ) or (x v x 2> x. 3 ) . Itf mathematical definition 
is a set of quantities (%, x z , x s ) taken in that order.* 
For any number of dimensions, therefore, a point is 
such a set of quantities as (x lt x 2 . . . x n ) taken in that 
order. A small w-dimensional region is thus an aggre- 
gate of such sets where all the %'s, x%s, . . . x n 's are 
nearly equal. 

Summary. Two-dimensional space is a region in 
which two independent co-ordinates only are required 
to define a point. It is necessarily superficial, but the 
surface may be either flat or curved. Space in which 
all the g's can be reduced to unity is called Euclidian 
space, whatever the number of dimensions. To genera- 
lize the notion of two-dimensional space curved in a 
third, curvature is defined by a certain mathematical 
expression, which is then generalized. The ideas of 
continuous and discontinuous curvature may be repre- 
sented in like manner by mathematical symbols, 
however many dimensions there may be under 
consideration. 

* Professor G. B. Mathews, F.R.S., " Nature," Vol. 106, February 
17, 1921, p. 290. 



CHAPTER XVIII 



THE GRAVITATION THEORY 



results of the last five chapters will now be 
Jl set down in parallel columns : 



GRAVITATIONAL FIELDS. 

(1) An artificial gravitational 
field can be destroyed, in other 
words transformed away, by 
changing the system of refer- 
ence appropriately. 

(2) A natural gravitational 
field cannot be transformed 
away wholly. 

(3) A natural gravitational 
field can be transformed away 
locally. 



(4) A local transformation 
distorts a natural gravitational 
field elsewhere. 

(5) A natural gravitational 
field cannot be transformed 
away where matter exists. 



GEOMETRY. 

(1) In Euclidian space the ^'s 
in the expression for a line 
element can be reduced to unity 
by a suitable transformation. 

(2) In non-Euclidian, or 
curved, space no transformation 
exists which will make all the 
^s unity everywhere. 

(3) Non-Euclidian space, if 
of continuous curvature, can be 
reduced locally to Euclidian 
space, the ^s becoming unity 
for a limited region. 

(4) Local reduction to Eucli- 
dian space distorts non-Euclidian 
space elsewhere. 

(5) Non-Euclidian space can- 
not be reduced to Euclidian 
space where discontinuous cur- 
vature occurs. 



As far as the illustrations in the previous chapters 
go, geometry and gravitation thus run on parallel lines, 

136 



THE GRAVITATION THEORY 137 

and this suggests that they may always do so. The 
illustrations raise the presumption that the g's which 
specify the geometry of a region also specify the gravita- 
tional forces in that region, whether natural or artificial, 
so that the five points set out above are not merely 
parallel but connected. The supposition includes, for 
example, the presumption that where the g's can be 
reduced to unity no gravitational forces other than 
artificial exist, and if the g's are reduced to unity by 
transformation, the same transformation ipso facto 
destroys the gravitational field : gravitational forces, 
if there are any, are necessarily artificial in a Euclidian 
region. In fact, geometry, as expressed by the g's 
is the exact counterpart of gravitation, and geometry 
and gravitation are but different aspects of the same 
thing. 

Stated in this way the supposition is too vague to 
enable, any deductions to be drawn from it. In order 
to obtain results which can be tested, it must be put 
into definite mathematical shape that is to say, 
embodied in one or more equations, and this is where 
the difficult and advanced mathematical work comes 
in. Einstein embodied his theory in a set of six 
equations, but it is not possible to give the work by 
which he obtained them, or even to state them. Little 
more can be done than to state the problem and the 
result. 

We carry forward from the restricted theory the 
notion of a four-dimensional continuum, and we have, 
therefore, from these equations to determine in terms 
of the co-ordinates the four g's in the relation 



138 THE THEORY OF RELATIVITY 

Here again the restricted theory helps by supplying 
the form which the expression for the interval element 
assumes for unaccelerated systems that is, for cases 
where there are no gravitational forces. This is the 
case in regions of a gravitational field so remote that 
the attracting matter which produces the field has no 
effect. Whatever values therefore are found for the 
g's they must be such as will reduce to the values 
i, i, i, and + i * at great distances from the 
attracting matter. The expression which Einstein 
obtains for the interval element, as the result of solving 
the six equations, is 

ds* = - - r z d0* - r 2 sin 2 6d<l>* + ydt* f . (10) 

where r, 6 and (/> correspond to the-polar co-ordinates 
explained in Chapter III (2), and 7 is written for 

s}/yyi 

brevity for i . m is the mass of the attractive 

particle to which the field is due. When r is very great, 

oyvr 

as it is in remote parts of the field, I , or 7, 

* It will probably be noticed that we have changed signs in the 
expression for ds* given in Chapter XI, writing ds* = - dx* - dy* - 
dz* + dP instead of ds* = dx* + dy* + dz* - df*. We have thus 
written - ds* for ds 1 . The sign given to ds* is a matter of conven- 
tion, and the first of the forms given above is preferable, having 
regard to the fact that df* is usually much larger than dx*> dy* or 
dte 2 , as will be seen later on in the chapter. The present conven- 
tion therefore keeps ds* essentially positive. 

t The reader who is unacquainted with trigonometry need not take 
any notice of the symbol sin? 6. In the subsequent work we shall 
adopt a simplification which will suppress it. It is desirable in the 
present instance to give the complete formula. 



THE GRAVITATION THEORY 139 

approaches unity.* If we put 7 = i in the relation 
(10) we get 

ds* = - dr* - r*d0* - r*sin*0d<l>* + dt 2 , 

which, by a transformation which the reader may take 
for granted, can be shown to be the same thing as 

^s 2 = - dx* - dy* - dz 2 + dP, 

in agreement with the restricted theory. 

If r, 6, </>, and t be identified respectively with %, 
# 2 > x 3 , and # 4 in the standard expression for the interval 
element, we see that 




2 _ 2m 

82- & = J ~ 

The reader does not need to be reminded that all this 
mathematical work is simply the statement of an hypo- 
thesis. The relation (10) results from the solution of 
a certain set of six equations which must be taken as 
a particular mathematical embodiment of the general 
supposition that the g's which determine the geometry 
of a region also determine the gravitational forces. 
These equations, which it has not been possible to give 
on account of their complexity, constitute the hypo- 
thetical law of gravitation which Einstein puts forward. 
He determines by means of them the values for the g's 
which have just been given, and he says that the g's 

* The same result follows if no matter is present anywhere, for 
then m = o, and y = i everywhere. 



140 THE THEORY OF RELATIVITY 

represent both the physical and the geometrical state 
of the gravitational field. This is nothing more than 
a plausible conjecture until it has passed the test of 
experiment, like every other hypothesis at the same 
stage. The experimental tests will be given in the 
next chapter. Meanwhile we proceed to make some 
further observations on the formula (10) in the sim- 
plified form 

n 



2m \ r 

r 

obtained by suppressing the space co-ordinate $, and 
thus reducing the conditions to two dimensions in space 
and one dimension in time. 

No transformation will reduce the g's in this expres- 
sion for the interval element to unity, consequently 
space in a gravitational field is non-Euclidian. It has 
a twist or curvature, and no figure obeying the theorem 
of Pythagoras can be drawn in it. We infer, therefore, 
that in our actual physical conditions Euclid's system 
is not exactly true that is to say, it does not exactly 
correspond to physical measurements. It might be 
thought that possibly this is due to the fact that the 
time dt (or dxj comes into interval measurements. 
This is not so, for if we measure up by a rod, both ends 
of the interval are necessarily measured simultaneously, 
and dt = o. The interval element then reduces to a 
line element given by 

ds* = - -- -- r*d0* 
2m 

J. "~~ * 

r 
and this cannot be changed into the form dxf + dxf 



THE GRAVITATION THEORY 141 

At distances remote from matter the expression (n), 
as we have seen, reduces to that of the restricted theory. 
Under such conditions all the g's are unity, but. they 
are not all of the same sign.* This characterizes what 
is called a semi-Euclidian continuum. If, however, 
measurements are made simultaneously, which can be 
effected by any observer in his own system, dt o, 
and he may regard his space as strictly Euclidian. 

Some further interesting conclusions may be drawn 
from the relation (n). Let us take an interval ds 
measured in the direction of r only, so that dd = o 
and dt o. We have then neglecting the negative 
sign 



or, ds = 



- zm/r 
dr 



2m /r. 

Again, take an equal interval and measure it perpen- 
dicularly to r, so that dr = o and dt = o. We then get 

ds * rdO. 

Since dt o in both cases, ds represents an actual 
length as measured by a rod. Now dr and rd0 are 
what these lengths should measure up to if they are 
to obey the relation 

ds 2 = dr* 



which is the condition for Euclidian space. But when 
measured along the radius vector, ds that is to say 
the length of the rod has to be multiplied by 

* See previous footnote. 



142 THE THEORY OF RELATIVITY 



xi 2mj r before it will fit into a Euclidian space. 
But x/i 2m /r is less than unity, and thus the rod 
contracts when placed radially. Transverse measure- 
ments require no change.* 

If r = 2m, i 2m/r o, and the length of the 
measuring rod vanishes. Thus, as we approach an 
attracting particle there comes a time when the length 
of the measuring rod vanishes and no advance is made, 
however many times we apply it.f This need not be 
taken as representing an actual physical happening. It 
is only the way our equations have of telling us that 
matter is impenetrable. Another effect of a similar 
kind concerning time will be considered in the next 
chapter. J 

We are now in a position to consider more closely the 
amount of indebtedness of the general theory to the 
restricted theory. In Chapter XII all we carried for- 
ward with us was a sense of dissatisfaction at the 
limitation to unaccelerated systems, and it was stated 
that the general theory was logically independent of 
the restricted theory. This is strictly true, but it is 
also true that the restricted theory has furnished several 
valuable data without which the development of the 
general theory would have been practically impossible. 
It is inconceivable that the six equations embodying 
the general law of gravitation could have been stated 

* Eddington, "Report," pp. 27 and 47. 

t Eddington, " Space, Time, and Gravitation," p. 98. 

I The reader will of course not confound these changes in lengths 
and times due to a gravitational field with those considered in con- 
nexion with the restricted theory, which occur in a non-gravitational 
field and are due to movement. 



THE GRAVITATION THEORY 143 

without reference to the restricted theory which led to 
the concept of the four-dimensional continuum. Nor 
would the particular solution which led to the expres- 
sion (10) for the line element have suggested itself had 
it not been clear that this solution must resolve itself 
into the form given in Chapter XL 

The idea, referred to in Chapter XI, of taking the 
velocity of light as unity, has also led to considerable 
simplification, though this cannot be made so evident 
here as in a more detailed work, but one curious con- 
sequence of this convention needs remark. It brings 
out the fact that progress through the four-dimensional 
space-time continuum is very much more rapid in the 
time direction than in the space directions. That is to 
say, unless a particle is moving with a velocity compar- 
able with that of light, the rate of change of its time 
co-ordinate with respect to any of its space co-ordinates 
is very great. For all ordinary objects, the " world- 
lines " are very nearly what we should call straight if 
we were speaking of progress through space only, and 
they are nearly parallel to the time axis. This will be 
clear from the fact that light travels at the rate of 
300,000 kilometres per second. If, therefore, we call 
this velocity unit velocity, one second of time is the 
equivalent of 300,000 kilometres. Whence the time 
unit, one second, laid off along the time axis is 300,000 
times the unit of length, or i kilometre, laid off any 
of the length axes. 

It is impossible to give any accurate picture of what 
is happening in a gravitational field, according to 
Einstein's view, but it still seems possible to get some 
general idea by the aid of analogies, though all analogies 



144 THE THEORY OF RELATIVITY 

are necessarily rough or even grotesque. The new 
theory concentrates attention upon the courses which 
objects pursue, the older theory upon forces which are 
supposed to influence bodies so as to make them follow 
these courses.* According to the Newtonian view, in 
the absence of force all bodies have a natural path in 
space, namely, a straight line described with uniform 
velocity. If, however, any other body be present it 
exercises a pull on the first body, drawing it out of its 
natural path if the two bodies are not in the line of the 
path, but accelerating its motion, in any case. A 
corresponding pull is exerted by the first body on the 
second. If the new point of view which Einstein 
invites us to adopt, presents a difficulty, it is useful to 
remember that the Newtonian view presented no less 
difficulty to philosophers in his day. Their great objec- 
tion was that it involved action at a distance, attri- 
buting to bodies a power to act where they are not. It 
seemed incredible that the sun acted across intervening 
space and pulled a planet out of the straight path 
which it would otherwise follow. Only the clearest 
evidence that this theory actually did give an explana- 
tion of the planetary motions, and presented a picture 
of what, in fact, went on in the solar system, surpassing 
by far in adequacy and accuracy any theory previously 
advanced, induced philosophers as a body to accept 
such action as possible. We have now grown so 
accustomed to it that nothing else seems natural to us. 
Einstein does not deny the influence of matter, but 
he gives us a different picture. It is as though space 

* Eddington, "Space, Time, and Gravitation," pp. 95, 96; cf. 
Chapter XIII (5). 



THE GRAVITATION THEORY 145 

were filled with some medium or substance matter we 
must not call it but some medium which acts as a 
guide to bodies passing through it. A solitary body is 
guided in a uniform straight course, or allowed to follow 
such a course ; it is immaterial which way we put it. 
But now introduce into the medium another body. 
Forthwith there is set up a twist, or strain, or curvature 
in the medium throughout space, intense near the body 
in proportion to its massiveness, and fading away and 
eventually disappearing in remoter parts, but every- 
where continuous without gaps or sudden changes, 
excepting in the places actually occupied by the bodies. 
The result of this action, which we figure as a change of 
structure of the medium, is to guide the first body into 
a curved path and to produce all the effects which we 
have grouped under the term acceleration. Of course, 
the first body distorts the medium in a similar way 
and sets up a corresponding disturbance in the motion 
of the second. This twist or curvature corresponds to 
what we have called the curvature of space. We notice 
that action at a distance is eliminated. The second 
body affects the medium in its immediate neighbour- 
hood, and this portion affects those in contact with it, 
and so on, just as a disturbance set up in water by a 
moving body is propagated outwards and affects other 
bodies in the same mass of water. 

The analogy just given is most imperfect, more 
especially because it is stated in terms of the ordinary 
three dimensions only. We have to suppose that all 
that has been described takes place in the four-dimen- 
sional space-time continuum, where pictorial diagrams 
fail and where the symbolical representations of mathe- 

10 



146 THE THEORY OF RELATIVITY 

matics are all we have to depend upon. But by sup- 
pressing one of the three spatial dimensions, some sort 
of a picture may be made even of this state of affairs. 
Consider a bundle of straight rubber tubes, and suppose 
progress in the direction of their length to represent 
duration in time, movement in any other direction 
being displacement in space, as usual. A small particle 
projected down one of the tubes will thus appear to be 
growing old, but will appear to be fixed spatially. Now 
let a massive body be projected down a tube somewhere 
in the middle of the bundle, and imagine that the effect 
is to twist the bundle so that the tubes present the 
appearance of the strands of a rope. The tube con- 
taining the particle will thus be twisted into a helix 
like a screw, and the particle will be constrained to 
follow its course. The projection of this course on any 
cross section of the bundle of tubes will be an oval or 
round curve, and if for " particle " we read " planet " 
and " sun ''for " massive body," we have a picture 
of a planet and its orbit. The tubes represent the 
world-lines of the bodies moving within them. In view 
of what has just been said in connexion with the unit 
value of the velocity of light, the helices which the 
tubes form will be so elongated axially as to be almost 
straight. 

Summary. The parallel between geometry and 
gravitation suggests that the g's which specify the one 
also specify the other. The embodiment in mathe- 
matical terms is a set of six equations, which con- 
stitute the law of gravitation. By means of these the 
g's are determined in a form consistent with the re- 



THE GRAVITATION THEORY 147 

stricted theory of relativity. Their values show that 
space is non-Euclidian in a gravitational field. In the 
absence of a gravitational field it is semi-Euclidian. In 
a gravitational field a measuring rod contracts when 
placed radially. Matter is mathematically expressed 
as a discontinuity. The general theory is logically 
independent of the restricted theory, but is stated so 
that the restricted theory is a particular or limiting 
case. World-lines are usually nearly " straight ". 
The Newtonian view concentrates attention upon 
forces acting at a distance causing bodies to pursue 
certain courses, while the new view concentrates upon 
the courses themselves. 



CHAPTER XIX 
THE CRUCIAL PHENOMENA 

THREE deductions which can be submitted to 
experimental tests have been made from the results 
of the last chapter. They are known as the " Crucial 
Phenomena," because they stand in relation to 
Einstein's theory as the necessary experimental com- 
plement which is required, as in the case of every 
hypothesis, before it can find acceptance. These 
phenomena relate to the following : 

(1) The motion of the apse of the planet Mercury. 

(2) The bending of light rays by the sun. 

(3) The displacement of lines in the Solar spectrum. 

i. THE ORBIT OF MERCURY. 

According to Newton's law, if the solar system con- 
sisted of two bodies only, the sun and a planet, the 
planet would describe round the sun an ellipse unvary- 
ing in shape, size, and position, and having the sun in 
one of its foci. The effect of the mutual attractions of 
the planets on one another, however, is to disturb the 
orbits in various ways. Amongst other things, each 
orbit rotates slowly, pivoted on the sun, while the planet 
revolves in it. 

Thus, if a planet be pictured as a bead sliding on an 

148 



THE CRUCIAL PHENOMENA 149 

oval wire while the wire itself keeps turning round in 
the same direction as that in which the bead slides, an 
idea of the actual path of a planet in space relatively 
to the sun may be obtained. Or, still better, cut out 
an ellipse from a piece of card and fasten it down on a 
table by a pin through the focus 5. Now take a pencil, 
and while rotating the card slowly and evenly about S, 
trace round the circumference of the card in the same 
direction, but more quickly. A pattern such as that 
shown in Fig. 36 will be obtained. The apses, which 




FIG. 36. 

is the name given to the pointed ends of the orbit, 
A, A' t A", etc., and P, P' ' , etc., where the planet is at 
its greatest and least distance from the sun, thus 
advance that is, they turn round in the same direction 
as the revolution of the planet. Each apse advances 
every year through an angle such as PSP'. The French 
astronomer Leverrier, in the survey of the solar system, 
in the course of which he was led to the discovery of 
the planet Neptune, calculated amongst other things 
the amount of advance of the apses of the planets 



ISO THE THEORY OF RELATIVITY 

according to Newton's law, and he found that the 
calculated amount agreed with the observed amount 
as nearly as could be expected in every case, excepting 
in that of Mercury. The apse of Mercury was observed 
to advance every year by considerably more than the 
calculated amount. Of course, this is comparatively 
speaking, for the actual amounts are exceedingly small. 
The figures are in seconds of arc per century : 

Observed advance . . . 574" 
Calculated advance . . . 532" 
Unexplained discrepancy . . 42" 

Were it not that Mercury has an exceptionally pointed 
orbit and a comparatively rapid motion, so that the 
cumulative effect can be observed with relative ease, 
this discrepancy would be too small to notice. Some- 
thing of the sort is doubtfully observable in the case of 
Mars, but in no other. 

Various suggestions have been made to account for 
this discordance. Leverrier himself thought that it 
was due to some undiscovered planet, as in the case of 
Uranus, but this time inside the disturbed orbit.* 
But no such planet has ever been found, and all other 
explanations have similarly failed until the time of 
Einstein. 

The actual work of finding an orbit is very similar 
whether we take Newton's or Einstein's law. In the 
latter case it is somewhat more complicated. But in 
principle the two points of view differ very materially. 
In the Newtonian case we suppose a planet, or particle 
as we may call it, to be started moving with a velocity 

*Eddington, "Space, Time, and Gravitation," p. 124, 



THE CRUCIAL PHENOMENA 151 

given in direction and magnitude at a stated place, 
and we find its path when some central body, the sun 
say, pulls on it with a force which varies according to 
Newton's law. If we adopt Einstein's way of looking 
at the matter we ask ourselves how the particle will 
move if projected with a given velocity as before, but 
otherwise moving freely, in so far as the curvature of 
space or the distortion due to the sun of the medium 
filling space will allow. The result of accepting the 
values of the forces deduced from the g's determined 
in the last chapter is to add a small quantity to Newton's 
statement, which accounts for the extra advance of 
Mercury's apse. Newton's law, though very nearly 
true, is only approximate. 

2. THE BENDING OF LIGHT RAYS 

It was shown in Chapters XIII and XIV that straight 
lines in non-gravitational fields were distorted into 
curves in artificial gravitational fields, and the same 
thing happens in natural gravitational fields. A ray of 
light is straight in vacuo in the absence of gravitation, 
and it may therefore be expected to become curved in 
the presence of matter. 

It has long been held by philosophers that light has 
mass. If this mass were subject to Newtonian gravita- 
tion, a ray of light passing near a heavy body such as 
the sun would follow a definite orbit and would be 
deflected. Einstein's theory, however, apart from the 
question of the mass of light, shows that the course of 
a ray depends definitely upon the geometry of the 
space through which it moves, and predicts an amount 



'52 



THE THEORY OF RELATIVITY 



of deflexion double of that which is to be expected 
from Newton's law. The only practicable test is to 
observe the apparent displacement of the fixed stars 
when light rays from them pass near the sun on their 
way to the earth. A star at T (Fig. 37) sends out rays 
in straight lines in all directions. One of these, TAB, 
strikes the earth E and renders the star visible ; other 




rays such as TA'E' miss the earth. Now interpose the 
sun 5 near the paths of the rays. The effect is to bend 
the ray TA (Fig. 38) towards the sun into the direction 
AE lt so that it now misses the earth. The ray which 
reaches the earth is the ray TA', which is now bent in 
the direction A'E. Thus the star is now seen at T' 
on EA' produced, instead of on ET as before, and the 




effect of the sun has been to displace its apparent 
position outwards. According to Einstein, this dis- 
placement should be about double the amount predicted 
by the older theory 1*74 seconds of arc as against '87 
of a second. These angles are very small (one second 
of arc is the angular diameter of a halfpenny over three 
miles away), and observations are possible only during 
a total eclipse, stars near the sun being lost in the sun's 



THE CRUCIAL PHENOMENA 153 

rays and therefore invisible at any other time. The 
effect was observed during the eclipse of the sun of 
May 29, 1919. Photographs taken of the region near 
the sun were compared with photographs of the same 
region when the sun was out of the way, and the 
difference showed the effect to the satisfaction of the 
astronomers, though many of the photographs were 
spoilt by cloud. Several attempts have been made to 
explain the results of observation by the assumption of 
a dense refracting atmosphere near the sun, but the 
deflexion has resisted all explanations other than that 
upon which Einstein based his prediction. 

3. DISPLACEMENT OF LINES IN THE SOLAR SPECTRUM 

According to the electron theory of matter atoms are 
very highly complicated structures composed of minute 
bodies charged with negative electricity called electrons, 
revolving round a central nucleus like planets round the 
sun. Their periods of revolution show remarkable 
constancy for the same substance, and their motion 
produces all the effects of electricity in violent oscilla- 
tion. They behave, in fact, like the oscillators in 
wireless telegraphy, and send out electro-magnetic 
waves in all directions. In wireless telegraphy these 
waves are large, but the waves set up by the electrons 
in an atom are excessively minute. When the atom 
is hot the disturbances which constitute the waves 
become intense, and those which are of the proper 
period become capable of affecting the eye with the 
sensation of light. The period remains constant for 
the same substance whatever the temperature, and 
consequently the electrons behave like remarkably 



iS4 THE THEORY OF RELATIVITY 

regular and efficient clocks. Now one of the things 
which Einstein showed was that the rates of clocks 
depend upon the value of g 4 , and they lose time.* 
The vibrations of all the electrons are slowed down, and 
it can further be shown that the more rapid vibrations 
are retarded more in proportion than the slower ones. 
The result is that the violet rays, which correspond to 
the more rapid vibrations, are affected to a greater 
extent than the slower ones, which produce the colour 
red, and thus the spectrum becomes crowded up towards 
the red when the source of the light is in a strong 
gravitational field. It was, therefore, expected that 
light rays issuing from any particular substance in the 
sun would be displaced towards the red, compared with 
those issuing from the same substance on earth. The 
observations are extraordinarily difficult, and the 
evidence of the existence of the effect is conflicting. 

The failure to detect this prediction of Einstein may 
be due either (i) to the extreme difficulty of the observa- 
tions ; (2) to the existence of some other effect which 
masks the displacement, but which is as yet undis- 
covered ; or (3) to the failure of Einstein's theory. 
This third alternative does not necessarily involve the 
untruth of the whole theory, but only its inapplicability 
to the phenomena of radiation, of which light is a 
particular case. The matter at stake is in fact the 
applicability of Einstein's theory to quantum pheno- 
mena. An outline of the quantum theory was given 
in Chapter II (4), and it is regarded as of the very 
highest importance by physicists. It is, in fact, of 

* Eddington, " Report," p. 56, 



THE CRUCIAL PHENOMENA 155 

the same order of importance as the atomic theory of 
matter, and it has found general acceptance. If, there- 
fore, it were once definitely shown that Einstein's 
theory failed in its applicability to this class of pheno- 
mena, its generality would be very seriously impaired. 
So far there does not appear to be any conclusive 
evidence one way or the other. 

The position therefore stands that Einstein has 
explained an outstanding difficulty in the Newtonian 
theory ; it has predicted an unlooked-for effect on light 
and explained it, while the truth of a second prediction 
is in doubt. The balance of experimental evidence is 
therefore so far in favour of the truth of the theory, 
but, in addition to this evidence, there are other 
considerations which are not strictly experimental, but 
which, taken together, tend in the same direction. 
These, however, will find a more fitting place in the 
next chapter. 

Summary. The experimental tests of the theory 
relate to (i) the orbit of Mercury ; (2) the bending of 
light rays in a gravitational field ; (3) the crowding up 
of spectral lines towards the red end. The first two 
have been verified, but not the third The importance 
of the third lies in its relation to the quantum theory. 



CHAPTER XX 
THE APPLICATION OF THE GENERAL PRINCIPLE 

THE principle of equivalence has thus led to the 
practical identification of geometry and gravita- 
tion, but this result, important and even impressive 
though it is, must not be allowed to obscure the main 
issue. It is, after all, only a means to an end, the end 
being the application of the General Principle of 
Relativity to the statement of physical laws. In order 
to complete the subject it has still to be shown how 
the gravitation theory enables these laws to be stated 
in identical forms, no matter what systems of reference 
are used. 

This work is still incomplete. Should the theory of 
relativity find general acceptance, it will occupy 
physicists for many years to come. Those who can 
understand the somewhat advanced mathematics which 
are required, will find an indication of the initial steps 
in Professor Eddington's report to the Physical Society 
on the Relativity Theory of Gravitation, to which 
frequent reference has been made in this book. The 
whole work will mean the complete re- writing of mathe- 
matical physics in the new terms. The mathematical 
limits of the present book prevent any attempt in this 
direction. In dealing with the restricted principle we 



APPLICATION OF THE GENERAL PRINCIPLE 157 

had to stop short at the application, and the same 
obstacle, exaggerated on account of the still more 
advanced mathematics required, hinders us here. The 
following observations must therefore be of a very 
general character. 

When phenomena are referred to the four-dimensional 
space-time continuum, it is found that the general facts 
of physics can be expressed in terms resembling the 
invariant expression for the interval element in that 
they do not alter their form, whatever reference system 
is used. Alteration of the reference system, as we have 
seen, introduces artificial forces, which are indistin- 
guishable from gravitational forces. The change of 
co-ordinates and the consequent change in the forces 
cancel one another out, so to speak, and preserve the 
form of the mathematical expressions which the change 
of co-ordinates might otherwise be expected to modify. 
It is as though the alteration in point of view brought 
into action an automatic governor, and switched in 
some agency which maintained the balance. The 
supreme importance of gravitation is thus manifest. 
Instead of being, as heretofore, a thing apart amongst 
natural agencies, it assumes, as it were, a controlling 
place. It is the counterpart of the geometry in terms 
of which all physical phenomena must be stated, and 
it corresponds with the circumstances of every observer 
so that he can make his statements in forms identical 
with those of any other. 

A very remarkable consequence follows from this. 
Since all reference systems are equivalent, they may 
change from time to time without any corresponding 
change occurring in statements of law. The changes 



158 THE THEORY OF RELATIVITY 

themselves produce the necessary balance. It matters 
not whether, according to the standards of A, B's 
system is rigid or not. All that can happen is that 
B's geometry will differ from that of A, and a corre- 
sponding difference will arise between their gravitational 
fields and produce automatic compensation. If B 
assumes his system to be rigid he necessarily assumes 
that A's standards are not, for it was by these standards 
that A judged the rigidity. Thus neither observer need 
attribute rigidity to the other's system or standards, 
though he assumes it for his own. The same con- 
siderations apply to the regularity of clocks. Every- 
body uses his own local measures, and all express the 
general results in identical form. 

This relieves us from the necessity of defining rigidity. 
Just as it is impossible to define position or motion 
without reference to some object, so we cannot define 
rigidity without reference to some body assumed to be 
rigid. This body again requires comparison with a 
third in order to test its rigidity, and so on without 
limit. We now see that no such definition is necessary. 
We define any length as so many multiples or sub- 
multiples of a standard unit, and this unit is the distance 
between two marks on a metal bar under specified 
physical conditions. This is our standard length, and 
no further trouble need be taken to ascertain whether 
it is rigid in the absolute sense or not. So also for 
clocks. We may take anything we like as the standard 
of time a rotation of the earth, the time of vibration 
of a sodium atom or any other convenient unit. 

The preceding chapters may have created the uncom- 
fortable impression on the reader's mind that he has 



APPLICATION OF THE GENERAL PRINCIPLE 159 

been led into the somewhat mystifying region called a 
four-dimensional space-time continuum and left there 
to extricate himself as best he can. The reflexion that 
this region can be represented adequately by mathe- 
matical symbols will probably not afford much relief. 
This feeling is not likely to oppress those who are 
accustomed to mathematical symbols, and who know, 
for example, the very practical physical results which 
follow from investigations involving imaginary expres- 
sions, such, for instance, as the square root of i. 
The difficulty in removing this impression, if it exists, 
lies in the very limited amount of mathematics which 
the writer is allowing himself. It seems to require a 
much smaller amount of mathematics to get into a 
four-dimensional continuum than to get out of it again. 
The reader may therefore be reminded that he has not 
been led into this region. What has been done is to 
point out that he and everything else were in it already, 
and always have been there. We may repeat the oft- 
quoted words of Minkowski : " The views of space and 
time, which I have set forth, have their foundation 
in experimental physics. Therein is their strength. 
Their tendency is revolutionary. From henceforth 
space in itself and time in itself sink into mere shadows, 
and only a kind of union of the two preserves an inde- 
pendent existence." * 

As a matter of fact, these investigations result in 
relations between space and time which are no more 
essentially mysterious than those to which everyone is 
accustomed in civil life, though the method by which 

* Eddington, "Space, Time, and Gravitation," p. 30. 



160 THE THEORY OF RELATIVITY 

the results were attained involves this unfamiliar 
combination of the two.* 

It has already been seen in Chapter XI that the 
physical history of every object is its world-line. The 
whole of physical nature in the mathematical diagram 
is a mass of these world-lines existing in the four- 
dimensional continuum like strings in a piece of jelly, 
and sometimes intersecting one another. The inter- 
sections of the world-lines of observers with other 
world-lines mark phenomena. The essential order of 
these intersections is not disturbed by distortion any 
more than the order of the intersections of the strings 
in the jelly, though it might so appear to particular 
individuals. It is this order which matters. It is 
independent of any individual point of view, which is 
the same thing as saying that the imposition of any 
particular reference system, or system of co-ordinates, 
makes no difference to it. The physical laws of nature 
which are stated in invariant fashion concern these 
world-lines, and the fact that they are independent of 
any particular co-ordinate system is therefore only 
what we should expect. 

The effect upon many, perhaps most, minds of the 
study of the application of the General Principle to 
Physics, is to create a strong bias in favour of the truth 
of the theory. This, of course, can hardly be said to 
be evidence, but when one sees familiar and well- 
established results coming out of it as well as new ones, 
an impression of coherence and unity is created which 
appeals to the artistic instincts if not to the strictly 

* E.g., the results in Chapter XIX. 



APPLICATION OF THE GENERAL PRINCIPLE 161 

scientific ones. Besides what is called its heuristic 
power the power of finding things out the theory 
seems to promise a unification of physical knowledge 
on a scale hitherto deemed impossible. It is the dream 
of some enthusiasts that it may be the means of 
unifying all knowledge, and that it may one day lead 
to the expression of all activities by a single equation. 
The writer has his doubts. He finds it difficult to 
fancy a sermon resolving itself into a blackboard 
demonstration with a differential equation as the text. 



Summary. The Gravitation Theory is a step in the 
application of the General Principle to the statement 
of Physical Laws. This application is in process of 
being worked out. Gravitation is now linked up with 
other physical agencies. The equivalence of reference 
systems renders definition of rigidity of length standards 
or regularity of time standards unnecessary. The 
results of inquiries conducted in terms of the four- 
dimensional continuum are expressed in the ordinary 
terms of three-dimensional space and one-dimensional 
time. Physical nature is made up of world-lines. The 
order of the intersections of world-lines is the important 
fact in nature, and this order is unaffected by any choice 
of reference systems. The heuristic value of the theory, 
and its power of unifying knowledge, create an impres- 
sion of its truth. 



ii 



CHAPTER XXI 
GENERAL SUMMARY AND CONCLUSION 

WE have now come to the end of the subject as 
defined by the title of the book and introduced 
the Theory of Relativity. There is nothing left but to 
summarize, and to add some final observations. 

After defining relativity as the theory of the state- 
ment of general physical laws so as to express them in 
identical forms, in spite of differences in the points of 
view of observers, the vague idea of a point of view 
was crystallized into the more precise concept of a 
reference system, a kind of framework fitted out with 
clocks, which is essential to the numerical statement of 
all phenomena, and which is, or may be, peculiar to 
every observer. Those comprehensive statements of 
fact called general physical laws were next considered. 
It appeared that these statements, since they all relate 
to measurement, must be expressed in mathematical 
terms, and that the subject matter of relativity relates 
to such expressions. It appeared further that an 
essential feature of these laws must be identity of 
form for different observers, since statements holding 
good for individuals, or small groups of observers only, 
cannot be called general, and are of no value as a means 
of putting facts on record for the benefit of others. 

162 



GENERAL SUMMARY AND CONCLUSION 163 

Physics, so far from being a coherent body of knowledge, 
would be but a Babel. But it was seen that by making 
use of unaccelerated rectangular systems, or Galilean 
systems as they were called, all mechanical laws could 
be stated in identical forms. Though space and time 
measurements might be made in units peculiar to each 
system, the general expressions comprehending the 
facts to which the measurements related, reduced to 
identical forms for all such systems, so that a general 
law applied everywhere though interpreted according 
to the several measurements of individuals. It was 
seen, however, that for such statements to be possible 
it was necessary to assume that measured lengths and 
times were not altered by relative movement between 
the systems ; that, for example, a yard measure on 
A's system meant to B the same as a yard on his own, 
and similarly with units of time, so that two observers 
could attribute the same length to the same object, or 
the same interval of time, notwithstanding their relative 
movement. If these assumptions were made, peculi- 
arities of individual systems, such as relative velocity, 
dropped out of account and mechanical laws showed no 
preference for one system over another. These assump- 
tions, in fact, made it possible to act upon the principle 
that all Galilean systems are equally suitable for 
the statement of general mechanical laws, and this 
statement was called the Mechanical Principle of 
Relativity. When, however, it was sought to extend 
the application of this principle from mechanical laws 
to electro-magnetic laws, electro-magnetic laws ap- 
peared to have a preference for a reference system at 
rest in the medium in which electro-magnetic agencies 



1 64 THE THEORY OF RELATIVITY 

operate. If the laws were stated in terms of a Galilean 
system moving with reference to this medium, the 
velocity of the system entered into the statements and 
so deprived them of generality. But it further appeared 
that if the suppositions regarding identity of measure- 
ments of lengths and times were abandoned and replaced 
by certain others, according to which the lengths of 
objects measured in the direction of motion, and 
measured times, did not appear the same to two 
observers in relative motion, electro-magnetic laws 
preserved their identity of form no matter to what 
Galilean system they were referred. These new sup- 
positions were, however, recognized as being more or 
less empirical, though supported by electro-magnetic 
considerations. Einstein showed that these new sup- 
positions could be derived from the remarkable fact 
that the velocity of light is the same relative to every 
observer. This fact makes the velocity of light unique 
amongst all other velocities, but it follows directly from 
the two postulates : 

(1) That no observer can detect his own motion 

through the medium which transmits light. 

(2) That the velocity of light in vacuo is independent 

of that of its source. 

The two new suppositions were thus put upon a basis 
independent of any electro-magnetic considerations. 
With their aid it was possible to extend the Mechanical 
Principle of Relativity and to act upon the principle 
that all Galilean reference systems are equally suitable 
for the statement of general physical laws. This is 
called the Restricted Principle of Relativity, because 
its operation is limited to Galilean reference systems. 



GENERAL SUMMARY AND CONCLUSION 165 

The new suppositions led to some remarkable con- 
clusions respecting estimates of lengths and times, 
velocities, masses, the simultaneousness of events, and 
so forth, and were embodied in a set of equations called 
the Lorentz transformation, by which physical events 
and geometrical statements in any Galilean system 
could be related to those in any other. It was further 
seen that the new suppositions necessitated modifica- 
tion of mechanical laws from their original form. 

It was seen that the invariant expression for a line' 
element contained as many terms as there were dimen- 
sions of space. The Lorentz transformation was 
applied to the three-dimensional expression, and the 
result showed that when this transformation is used, 
an invariant expression must contain four terms, a 
time term being a necessary addition. The concept 
of objects as four-dimensional, and existing in a four- 
dimensional continuum, is thus the necessary conse- 
quence of the use of the Lorentz transformation. 
Though no actual picture can be formed corresponding 
to this concept, it can be represented adequately by 
mathematical symbolism. 

No logical reason being assignable for restricting the 
statements of physical laws to unaccelerated systems, 
the conditions were examined under which it might be 
possible to express them in terms of Gaussian systems. 
It was seen that the forces which were necessarily intro- 
duced form an obstacle which could, however, be over- 
come by the adoption of the Principle of Equivalence. 
An examination of the principle of equivalence brought 
out a parallel between gravitational fields of force and 
their geometry, which led to Einstein's supposition that 



1 66 THE THEORY OF RELATIVITY 

the quantities which enter as multipliers into the 
expression for the interval element also specify the 
gravitational forces in any region. This supposition 
can be formulated definitely in a set of six equations 
which constitute Einstein's hypothetical law of gravita- 
tion, and a solution of these can be obtained, advantage 
being taken of the fact that the expression for the inter- 
val element must reduce to the restricted theory form. 
This solution shows that when matter is present space 
is non-Euclidian. Einstein proposed three deductions 
from his hypothesis as crucial tests, and two of these 
have been confirmed. The automatic introduction of 
gravitational forces consequent upon change of co-ordi- 
nates enables the form of expressions of physical laws, 
when stated in terms of the four-dimensional space-time 
continuum, to be preserved. The application of the 
principle that all Gaussian four-dimensional co-ordinate 
systems are equivalent for the statement of general 
physical laws is thus made possible. This principle is 
called the General Principle of Relativity. 

THE /ETHER 

In the previous pages no special pains have been 
taken to draw any clear distinction between space and 
the medium called the aether, which philosophers have 
supposed to pervade space, and which serves as the 
vehicle for the transmission of light and other electro- 
magnetic radiations, and generally as the seat of 
actions not ascribable to matter. No distinction 
seemed necessary, for as long as it was realized that 
light was conveyed and that curvature or twisting 
existed, questions as to what conveyed the light, or 



GENERAL SUMMARY AND CONCLUSION 167 

what it was that was twisted or curved, did not seem 
to be of immediate relevance. This omission, however, 
was a matter of convenience. Something further will 
now be said on the subject. 

It is very widely held that facts are against the exist- 
ence of the aether, and that Einstein's theory dispenses 
with the need of it. As far as the writer understands 
the matter, it is argued that because one set of experi- 
ences require a fixed aether, while another set fail to 
detect movement through it, it can neither be moving 
nor fixed, and that therefore it cannot exist. It seems, 
however, to the writer that those who, on the one hand, 
are lamenting the death of the aether, and on the other 
are executing war dances over its corpse, are over hasty. 
If by aether is meant something which has properties 
such as mass, impenetrability, rigidity, or elasticity, 
which are usually associated with matter, then, indeed, 
the .aether is dead. But if there is no medium of any 
kind, and nothing in space, we are compelled, so it 
seems to the writer, to attribute to empty vacuity the 
properties of transmitting light, and of assuming 
geometrical structure, which is very like a contradiction 
in terms, if not actually so. It is impossible to accept 
the supposition that nothing can do anything, even 
transmit light waves. 

Now, though it may be difficult to conceive of any- 
thing which has none of the ordinary properties of 
matter, but is yet capable of the activities which have 
just been named, there is no actual contradiction 
involved. It is the softer horn of the dilemma. There 
is, of course, the third alternative that aether and space 
are one and the same, but this seems to require us to 



1 68 THE THEORY OF RELATIVITY 

believe that there must of necessity be something in 
the interval between two bodies which, as it were, 
props them apart, and which, if removed, would cause 
or allow the interval to collapse. There is no evidence 
for any such supposition. There is nothing to show 
that nature abhors a vacuum to the extent of making 
two bodies coalesce. If, therefore, we are, as it would 
seem, driven to accept the fact that the universe of 
experience is filled with some entity which cannot be 
called matter, it is merely a question of words whether 
we call this entity aether or not. 

The fact, as it appears to the writer, is that so far 
from Einstein having destroyed the aether or rendered 
it superfluous, he has discovered in it the capacity of 
assuming some sort of geometrical structure in time- 
space. It is nothing new to think of the aether as 
subject to strain and thus capable of exhibiting geo- 
metrical properties, but this fresh capacity seems to be 
something of quite a different order. It may very 
well be that this is the beginning of the discovery of a 
series of properties which use and time may eventually 
weld into one concept, and that the aether, so far from 
being dead, is in process of being born. 

ACTION AT A DISTANCE 

The immediate predecessor of Newton's theory was 
the Cartesian theory of Vortices. According to this 
theory, space is rilled with a subtle medium or aether 
which is in a continual state of whirl, producing vor- 
tices which entangle bodies such as the planets and thus 
cause them to revolve. The theory of vortices broke 
down under analysis, but it was held to possess an 



GENERAL SUMMARY AND CONCLUSION 169 

advantage over Newton's theory in that the cause of 
the motion was present where the effect occurred, while, 
according to Newton's theory, the cause resided in a 
distant body. Newton's theory, therefore, required a 
body to act where it was not, and this was held to be 
inconceivable. 

As a way out of the difficulty, it has been suggested 
that since the sun manifestly acts upon the planets, 
and if action at a distance is impossible, then the sun 
must, in a sense, be present where the planets are. 
This suggestion, fantastic though it is, at least serves 
to show how strongly the idea of efficiency or adequacy, 
as part of the concept of cause, has impressed men's 
minds. It has been urged, if not as an argument in 
favour of Einstein's theory, at least as one of its advan- 
tages, that it dispenses with the idea of action at a 
distance. It does not assume, as the theory of Descartes 
seems to assume, that motion is caused or maintained 
by aether, but it implies that motion is in some sense 
determined or guided by an agency present where the 
body is, and acting directly upon it by contact. It 
does not give an explanation, any more than Newton's 
theory, of the agency which started the body moving, 
but it is held that it gives an intelligible picture of its 
subsequent movement, in which particular Newton's 
theory is thought to fail. 

While holding that efficiency is a proper and necessary 
part of the concept of cause in the philosophical sense 
of the term, the writer is unable to see that the theories 
of either Einstein or Descartes offer any advantage in 
this respect over Newton's theory. He is quite unable 
to understand how or why contact or collision between 



170 THE THEORY OF RELATIVITY 

bodies modifies their motion. It is a known fact that 
it does do so, but so does distant action at least, so 
we have become accustomed to think since the time of 
Newton, but the mechanism which produces this effect 
is just as mysterious in the one case as in the other. 
It is all the more mysterious where, as in the present 
case, the action occurs between a material body and a 
subtle substance like the aether, which is held on other 
grounds not to be impenetrable. Even when two 
material bodies collide, and it might be held that im- 
penetrability obliges one or other to give way, it still 
remains to explain impenetrability. This seems, to the 
writer, to be a most obscure property. If, as we now 
believe, matter is made up of atoms separated by great 
distances, each atom being composed of electrons 
separated by distances which are enormous relatively 
to their size, it is an extraordinary fact that bodies 
cannot pass through one another without action upon 
either. Electric fields may be invoked to explain it, 
but this is only action at a distance over again. 

THE LIMITED UNIVERSE 

The curvature of space, or of the aether, leads to the 
conclusion that any region, if sufficiently extended, 
may eventually bend round into itself, and thus that 
the universe of experience may be limited. Indeed, 
calculations have been made as to its dimensions, the 
amount of matter in it, and so forth. This does not 
necessarily mean that the universe is bounded. For, 
consider two-dimensional beings on the surface of a 
sphere. Their universe is limited to the surface, but 



GENERAL SUMMARY AND CONCLUSION 171 

they can wander about it freely without encountering 
a boundary. 

The use of the term universe in this connexion is 
somewhat unfortunate. It is open to the construction 
that nothing can possibly exist outside a limited region. 
All that can be meant is that there are geometrical 
limits to man's experience, and this, if true, is a highly 
important addition to knowledge. If there is anything 
bigger than this " universe," or if there is more matter 
anywhere, it cannot come within our knowledge, just 
as no velocity greater than that of light is measurable. 
The statement may be nothing more than the mathe- 
matical expression of the imprisonment of mankind in 
the present state of existence. 

PHILOSOPHY 

The separation of mathematics and physics from 
metaphysics, explained in Chapter II, is a matter of 
method only, and must not be held to imply that 
metaphysics is thereby ruled out of account as a serious 
subject of inquiry. Some such procedure had to be 
adopted if any progress were to be made in knowledge, 
owing to the failure of metaphysics to reach positive 
conclusions. It is an interesting matter for speculation 
whether the ancients or their successors would have 
thought it worth while to devote so much energy to 
philosophic speculation if they had grasped the possi- 
bilities of the method of hypothesis backed by experi- 
ment. It is, in the writer's belief, fortunate that they 
did not. They might have been diverted from philo- 
sophical inquiries to such an extent as to allow it to be 



172 THE, THEORY OF RELATIVITY 

forgotten that there was anything in such concepts as 
being, cause, space or time, other than those parts 
separated out for treatment by the mathematicians 
and physicists. It is not wrong to make this separa- 
tion ; it is a matter of necessity, and no error is 
imported by it into mathematics or physics. No error, 
for example, arises in physics from ignoring efficiency 
as part of the concept of cause, and denning cause as a 
necessary antecedent. It may be a great deal more, 
but it certainly is that, and whatever else it may be, 
does not concern the physicist as such. But though 
no error is entailed upon physics by this limitation, very 
Serious error might be entailed upon human thought 
by forgetting such matters as that cause might imply 
very much besides invariable antecedence. Philo- 
sophical speculation, barren though it has been in 
positive results, has played an important part in keeping 
to the front the belief that there may be other things 
in the universe besides the material. Einstein's theory 
points in the same direction. The remarkable feature 
about it is that starting from a purely experimental 
basis, it compels us to accept the supersensual as a fact. 
If the theory is true this conclusion seems inevitable. 



BIBLIOGRAPHICAL NOTE 

The literature of Relativity is already large. A 
bibliography appeared in " Nature," No 2677, Vol. 106, 
p. 811, and another is given in Dr. Slosson's book 
mentioned below. The following is a list of some of 
the works and papers bearing on the subject. They 
are in English unless otherwise stated. 

Vi. BIRD, J. MALCOLM. Einstein's Theories of Relativity and 
Gravitation. (Scientific American Publishing Co., Munn & Co., 
New York ; Methuen, London.) 

2. CARMICHAEL, R. D. The Theory of Relativity. (John 
Wiley & Sons, New York ; Chapman & Hall, London.) 

3. CARR, H. WILDON. The General Principle of Relativity. 
(Macmillan.) 

4. CHRISTOFFEL, E. B. Ueber die Transformation der 
homogenen Differentialausdriicke Zweiter Grades. (German.) 
Crelle's Journal, Vol. 70 (1869), pp. 46-70. See also Riemann 
and Dedekind referred to in this paper. 

v V5- CUNNINGHAM, E. The Principle of Relativity. (Cam- 
bridge University Press.) 

6. CUNNINGHAM, E. Relativity, the Electron Theory and 
Gravitation. (Longmans, Green & Co.) 

7. EDDINGTON, A. S. Report on the Relativity Theory of 
Gravitation. (Physical Society of London, Fleetway Press.) 

V Vs. EDDINGTON, A. S. Space, Time, and Gravitation. 

(Cambridge University Press.) 
v >/9- EINSTEIN, A. Relativity, the Special and General Theory. 

(Methuen.) 

10. FREUNDLICH, E. The Foundations of Einstein's Theory 

of Gravitation. (Cambridge University Press.) 



174 THE THEORY OF RELATIVITY 

V 

n. GAUSS, K. F. Disquisitiones Generates circa superficies 

curvas. (Latin.) Royal Society of Gottingen, October 8, 
1827. English Translation by J. F. Morehead and A. M. 
Hiltebeitel. (Princetown University.) 

12. HALDANE, Viscount. The Reign of Relativity. (John 
Murray.) 

^ 13. LORENTZ, H. A., EINSTEIN, A., MINKOWSKI, H. Das 
Relativitdtsprinzip . (A collection of papers in German ; 
Teubner, Leipsig, and Berlin.) 

v / ^4. RICHARDSON, O. W. The Electron Theory of Matter. 
(Cambridge University Press.) 

V ^15. RICCI, G., and -LEVi-CiviTA. I., Methodes de calcul 
differential absolu et leurs applications. (French.) Mathe- 
matische Annalen, Vol. 54 (1901), pp. 125-201. 

1 6. SAHA, M. N., and BOSE, S. N. The Principle of Rela- 
tivity. (A translation of the two main papers by Einstein in 
13 above, and of another paper with its appendix by Minkowski, 
together with notes and other matter. University of Calcutta.) 

Viy. SCHLICK, M. Space and Time in Contemporary Physics. 
(Clarendon Press.) 

>/i8. SILBERSTEIN, L. The Theory of Relativity. (Mac- 
millan.) 

Vig. SLOSSON, EDWIN E. Easy Lessons in Einstein. (Har- 
court, Brace, and Howe, New York.) 

N fV$o. WEYL, H. Space Time Matter. Translated by 
Henry L. Brose, Christ Church, Oxford. (Methuen.) 



INDEX 



ABSCISSA, 20. 

Acceleration, 32, 36, 38, 78, 

102, 109, no, 113. 
Action at a distance, 144, 145, 

168. 

^Ether, 15, 48, 52, 166, 170. 
Atoms, 153, 170. 
Axes, 20. 

CAUSE, 36, 169, 172. 
Circular measure, 122. 
Clocks, 12, 26, 68, 71. 
Continuity, 15, 25, 117, 134. 
Continuum, 8, 92, 94. 
Co-ordinates 

Cartesian, 19. 

Definition of, 20, 22. 

Galactic, 29. 

Galilean, 40, 97, 163. 

Gaussian, 25, 97, 98, 165. 

Polar, 23, 97, 121. 
Correlativity, 43. 
Curvature 

Definition of, 86, 134. 

Discontinuous, 127, 134. 

Spatial, 119, 129, 134, 136, 
145, 170. 

DECLINATION, 28. 
Descartes, 19, 168. 
Dimensions, 6, 88, 84, 89, 92, 
H9, 133- 



ECLIPTIC, 29. 

Electro-magnetic laws, 47. 
Electrons, 7, 83, 153. 



Element 

Definition of, 86. 

Line, 88, 121, 126, 165. 
Equivalence, principle of, 98, 

99, 1 1.6, 156, 165. 
Euclidian space, 131, 134, 136, 

141. 

Event, 26, 91, 94. 
Experiment, 13, 35. 
Explanation, 36. 

FIELD of force, 19. 
First point of Aries, 28. 
Flatness, 120. 

GALILEO, 40. 
Gauss, 25, 133. 
Geometry 

Plane, 123, 129. 

Specification of, 129. 

Spherical, 124. 
Gravitation 

Artificial, 98, 108, 112, 116, 

136, 137- 
Effects on measurements, 

141, 154. 
Einstein's law of, 117, 137, 

144. 

Features of, 103, 136. 
Importance of, 100, 157. 
Natural, 116, 136. 
Newtonian, 35, 144, 148. 



HYPOTHESIS, method of, 35, 
148, 171. 

175 



i 7 6 



THE THEORY OF RELATIVITY 



INERTIA, 32, 101, 106, 112. 
Initial line, 23. 
Interval, 91, 138. 
Invariant, 86, 157, 165. 

LAPLACE, 35. 

Latitude and longitude, 24, 28, 
29, 124. 

Leverrier, 149, 150. 

Light- 
Rays, curvature of, 151. 
Unitary value of velocity, 

9i, 143- 

Wave theory, 55, 167. 
Lorentz transformation, four- 
dimensional implication of, 
91. 

MAPS, 17. 
Mars, 150. 

Mass, 33, 79, 102, 106, 107, 112. 
Mathematical physics, 14, 156. 
Mathematics, 13. 
Mercury, 35, 148. 
Metaphysics, 10, 34, 76, 171. 
Minkowski, 159. 
Momentum, 33, 37, 80. 
Motion 

Absolute, 51. 

Definition of, 33, 37. 

First law of, 101. 

Second law of, 37, 40. 

NEPTUNE, 149. 
Newton, 35, 144, 148. 

ORDINATE, 20. 
Origin, 20. 

PARTICLE, 41. 
Physical laws 

Essentially mathematical, 

viii., i, 37. 

Generality of, 38, 97. 
Subject matter of, i, 34, 39, 
162. 



Physics, ii. 
Planck, 14. 
Point, 26, 135. 

event, 26. 

of view, 2, 9, 17, 19, 29, 
106, 162. 

Pole, 23, 29. 

Pythagoras, theorem of, 131. 

QUANTUM theory, 14, 154. 

RADIATION, 14, 52. 
Radius vector, 23, 
Reference frame, 18, 19, 28. 

system, 27, 32, 40, 162. 
Region, 135. 
Relativity 

Assumptions of mechanical 

principle, 43, 47, 64, 163. 
Heuristic value of, 161. 
Mechanics, 71. 
Of knowledge, 2. 

motion, 51. 

space and time measure- 
ments, 6, 49, 59, 66, 67, 92. 

Principles of, stated, 43, 51, 

69, 98, 163, 164. 
Relation between restricted 
and general theories, 70, 
96, 142. 
Subject matter of, vi. 2, 29, 

43. 7, 96, 162. 
Right ascension, 28. 
Rigidity, n, 12, 18, 157. 
Rotation, 32, 101. 

SEMI-EUCLIDIAN continuum, 

141. 

Simultaneousness, 72, 73. 
Space, physical, n. 
Spectral lines, displacement of, 

153- 

Speed, 31. 
Synchronism, 27. 
System, observer's, 27. 



INDEX 



177 



TIME, physical, n. 
Transformation- 
Definition of, 29, 45. 
Effects of, 84, 91, 98, 1 06, 

112, 115, 123, 136. 

Galilean formulae, 46. 
Lorentz formulae, 48, 67. 
Truth, 13. 


UNIVERSE, 170. 
Uranus, 150. 

VECTORIAL angle, 23. 
Velocity, 31, 74, ?8. 
Vortices, 168. 

WEIGHT, 33, 38. 
World-lines, 94, I43> 



> I 4 6 l6 - 



PRINTED IN GREAT BRITAIN AT THE UNIVERSITY PRESS, ABERDEEN 



A SELECTION FROM 

MESSRS. METHUEN'S 
PUBLICATIONS 

This Catalogue contains only a selection of the more important books 
published by Messrs. Methuen. A complete catalogue of their publications 
may be obtained on application. 



Armstrong (W. W.). THE ART OF 
CRICKET. Cr. 8vo. 6s. net. 

Bain (F. W.) 

A DIGIT OF THE MOON : A Hindoo Love 
Story. THE DESCENT OF THE SUN : A 
Cycle of Birth. A HEIFER OF THE DAWN. 
IN THE GREAT GOD'S HAIR. A DRAUGHT 
OF THE BLUE. AN ESSENCE OF THE DUSK. 
AN INCARNATION OF THE SNOW. A MINE 
OF FAULTS. THE ASHES OF A GOD. 
BUBBLES OF THE FOAM. A SYRUP OF THE 
BEES. THE LIVERY OF EVE. THE SUB- 
STANCE OF A DREAM. All Fcap. 8vo. 55. 
net. AN ECHO OF THE SPHERES. Wide 
Demy. 125. 6d. net. 

Baker (C. H. Collins). CROME. Illus- 
trated. Quarto. 5 55. net. 

Balfour (Sir Graham). THE LIFE OF 
ROBERT LOUIS STEVENSON. Twen- 
tieth Edition. In one Volume. Cr. 8vo. 
Buckram, 75. 6d. net. 

Bateman (H. M.). A BOOK OF DRAW- 
INGS. Fifth Edition. Royal 4/0 
IDS. 6d. net. 

SUBURBIA. Demy \io. 6s. net. 

Bell (Mary I. M.). A SHORT HISTORY 
OF THE PAPACY. Demy 8vo. 215. net. 

Belloc (H.) 

PARIS, 8s. 6d. net. HILLS AND THE SEA, 6s. 
net. ON NOTHING AND KINDRED SUB- 
JECTS, 6s. net. ON EVERYTHING, 6s. net. 
ON SOMETHING, 6s. net. FIRST AND LAST, 6s. 
net. THIS AND THAT AND THE OTHER, 6s. 
net. MARIE ANTOINETTE, i8s. net. 

Blackmore (S. Powell). LAWN TENNIS 
UP-TO-DATE. Illustrated. Demy 8vo. 
I2S. 6d. net. 

Carpenter (G. H.). INSECT TRANSFOR- 
MATION. Demy &vo. 125. 6d. net. 

Chandler (Arthur), D.D., late Lord Bishop of 
Bloemfontein 

ARA CCELI : An Essay inMystical Theology, 
55. net. FAITH AND EXPERIENCE, 55. net. 
THE CULT OF THE PASSING MOMENT, 6s. 
net. THE ENGLISH CHURCH AND RE- 
UNION, 55. net. SCALA MUNDI, 45. 6d. net. 

Chesterton (G. K.) 

THE BALLAD OF THE WHITE HORSE. ALL 
THINGS CONSIDERED. TREMENDOUS 
TRIFLES. ALARMS AND DISCURSIONS. A 
MISCELLANY OF MEN. THE USES OF 
DIVERSITY. All Fcap. 8vo. 6s. net. 
WINE, WATER, AND SONG, Fcap. 8vo. 
IS. 6d. net. 



Clutton-Broek (A.). WHAT IS THE KING- 
DOM OF HEAVEN? Fifth Edition. 

Fcap. 8vo. 55. net. 
ESS AYS ON ART. Second Edition. Fcap. 

8vo. 55. net. 
ESSAYS ON BOOKS. Third Edition 

Fcap. 8vo. 6s. net. 
MORE ESSAYS ON BOOKS. Fcap. 8vo. 

6s. net. 
SHAKESPEARE'S HAMLET. Fcap. 8vo. 

5s. net. 
Conrad (Joseph). THE MIRROR OF 

THE SEA : Memories and Impressions. 

Fourth Edition. Fcap. 8vo. 6s. net. 
Drever (James). THE PSYCHOLOGY OF 

EVERYDAY LIFE. Cr. 8vo. 6s. net. 
THE PSYCHOLOGY OF INDUSTRY. 

Cr. 8vo. 55. net. 
vEinstein (A.). RELATIVITY: THE 

SPECIAL AND THE GENERAL 

THEORY. Translated by ROBERT W. 

LAWSON. Seventh Edition. Cr.8vo. 5s.net. 
SIDELIGHTS ON RELATIVITY. Two 

Lectures by ALBERT EINSTEIN. Cr. 8vo. 

35. 6d. net. 

Other Books on the Einstein Theory. 
'SPACE TIME MATTER. By HERMANN 

WEYL. Demy 8vo. 215. net. 
EINSTEIN THE SEARCHER : His WORK 

EXPLAINED IN DIALOGUES WITH ElNSTEIN. 

By ALEXANDER MOSZKOWSKI. Demy 

8vo. i2s. 6d. net. 
AN INTRODUCTION TO THE THEORY 

OF RELATIVITY. By LYNDON BOLTON. 
, Cr. 8vo. 55. net. 

RELATIVITY AND GRAVITATION. By 

Various Writers. Edited by J. MALCOLM 

BIRD. Cr. 8vo. 75. 6d. net. 
RELATIVITY AND THE UNIVERSE. 

By DR. HARRY SCHMIDT. Second Edition. 

Cr. 8vo. ss. net. 
VtHE IDEAS OF EINSTEIN'S THEORY. 

By J. H. THIRRING. Cr. 8vo. 55. net. 
RELATIVITY FOR ALL. By HERBERT 

DINGLE. Fcap. 8vo. 2$. net. 
Evans (Joan). ENGLISH JEWELLERY. 

Royal tfo. 2 125. 6d. net. 
Fyleman (Rose). FAIRIES AND CHIM- 
NEYS. Fcap. 8vo. jfwelfth Edition. 

35. 6d. net. 
THE FAIRY GREEN. Sixth Edition. 

Fcap. 8vo. 35. 6d. net. 
THE FAIRY' FLUTE. Second Edition. 

Fcap. 8vo. 3$. 6d. ne}. 



MESSRS. METHUEN'S PUBLICATIONS 



Selous (Edmund) 

TOMMY SMITH'S ANIMALS. TOMMY 
SMITH'S OTHER ANIMALS. TOMMY SMITH 
AT THE Zoo. TOMMY SMITH AGAIN AT 
THE Zoo. Each 2S. gd. JACK'S INSECTS, 
35. 6d. JACK'S OTHER INSECTS, 35. 6d. 

Shelley (Percy Byssbe). POEMS. With 
an Introduction by A. CLUTTON-BROCK 
and Notes by C. D. LOCOCK. Two 
Volumes. Demy 8vo. i is. net. 

Smith (Adam). THE WEALTH OF 
NATIONS. Edited by EDWIN CANNAW. 
Two Volumes. Third Edition. Demy 
8vo. i los. net. 

Smith (S. C. Kalnes). LOOKING AT 
PICTURES. Ilustrated. Second Edi- 
tion. Fcap. 8vo. 6s. net. 

Spens (Janet). ELIZABETHAN DRAMA. 
Cr. 8vo. 6s. net. 

Stevenson (R. L.). THE LETTERS OF 
ROBERT LOUIS STEVENSON. Edited 
by Sir SIDNEY COLVIN. A New Re- 
arranged Edition in four volumes. Fourth 
Edition. Fcap. 8vo. Each 6s. net. 

Surtees (R. S.) 

HANDLEY CROSS, 7s. 6d. net. Mr. 
SPONGE'S SPORTING TOUR, 7s. 6d. net. 
ASK MAMMA : or, The Richest Commoner 
in England, 7s. 6d. net. JORROCKS'S 
JAUNTS AND JOLLITIES, 6s. net. MR. 
FACEY ROMFORD'S HOUNDS, 7s. 6d. net. 
HAWBUCK GRANGE ; or, The Sporting 
Adventures of Thomas Scott, Esq., 6s. 
net. PLAIN OR RINGLETS ? 75. 6d. net. 
HILLINGDON HALL, 75. 6d. net. 

Tilden (W. T.). THE ART OF LAWN 
TENNIS. Illustrated. Fourth Edition. 
Cr. 8vo. 6s. net. 

Tlleston (Mary W.). DAILY STRENGTH 
FOR DAILY NEEDS. Twenty-seventh 
Edition. Medium i6mo. 35. 6d. net. 



Turner (W. J.). MUSIC AND LIFE. 
Cr. 8vo. 7s. 6d. net. 

Underbill (Evelyn). MYSTICISM. A 
Study in the Nature and Development of 
Man's Spiritual Consciousness. Ninth 
Edition. Demy 8vo. 15$. net. 

Vardon (Harry). HOW TO PLAY GOLF. 
Illustrated. Fifteenth Edition. Cr. 8vo. 
55. 6d. net. 

Wade (G. W.). NEW TESTAMENT 
HISTORY. Demy 8vo. i8s. net. 

Waterhouse (Elizabeth). A LITTLE BOOK 
OF LIFE AND DEATH. Twenty-first 
Edition. Small Pott 8vo. as. 6d. net. 

Wells (J.). A SHORT HISTORY OF 
ROME. Eighteenth Edition. With 3 
Maps. Cr. 8vo. 5$. 

Wilde (Oscar). THE WORKS OF OSCAR 
WILDE. Fcap. 8vo. Each 6s. 6d. net. 
i. LORD ARTHUR SAVILE'S CRIME AND 
THE PORTRAIT OF MR. W. H. n. THE 
DUCHESS OF PADUA, m. POEMS, iv. 
LADY WINDERMERE'S FAN. v. A WOMAN 
OF No IMPORTANCE, vi. AN IDEAL HUS- 
BAND, vn. THE IMPORTANCE OF BEING 
EARNEST, vm. A HOUSE OF POME- 
GRANATES, ix. INTENTIONS, x. DE PRO- 

FUNDIS AND PRISON LETTERS. XI. Es- 

SAYS. xii. SALOME, A FLORENTINE 
TRAGEDY, and LA SAINTS COURTISANE. 
xin. A CRITIC IN PALL MALL. xiv. 
SELECTED PROSE OF OSCAR WILDB. 
xv. ART AND DECORATION. 

A HOUSE OF POMEGRANATES. Illus- 
trated. Cr. 4*0. 2 is. net. 

Yeats (W. B.). A BOOK OF IRISH 
VERSE. Fourth Edition. Cr. 8vo. 7*. 
net. 



PART II. A SELECTION OF SERIES 
The Antiquary's Books 

Demy 8vo. IDS. 6d. net each volume 
With Numerous Illustrations 



ANCIENT PAINTED GLASS IN ENGLAND. 
ARCHAEOLOGY AND FALSE ANTIQUITIES. 
THE BELLS OF ENGLAND. THE BRASSES 
OF ENGLAND. THE CASTLES AND WALLED 
TOWNS OF ENGLAND. CELTIC ART IN 
PAGAN AND CHRISTIAN TIMES. CHURCH- 
WARDENS' ACCOUNTS. THE DOMESDAY 
INQUEST. ENGLISH CHURCH FURNITURE. 
ENGLISH COSTUME. ENGLISH MONASTIC 
LIFE. ENGLISH SEALS. FOLK-LORE AS 
AN HISTORICAL SCIENCE. THE GILDS AND 
COMPANIES OF LONDON. THE HERMITS 
AND ANCHORITES OF ENGLAND. TH 



MANOR AND MANORIAL RECORDS. THE 
MEDIEVAL HOSPITALS OF ENGLAND. 
OLD ENGLISH INSTRUMENTS OF Music. 
OLD ENGLISH LIBRARIES. OLD SERVICE 
BOOKS OF THE ENGLISH CHURCH. PARISH 
LIFE IN MEDIAEVAL ENGLAND. THE 
PARISH REGISTERS OF ENGLAND. RE- 
MAINS OF THE PREHISTORIC AGE IN ENG- 
LAND. THE ROMAN ERA IN BRITAIN. 
ROMANO-BRITISH BUILDINGS AND EARTH- 
WORKS. THE ROYAL FORESTS OF ENG- 
LAND. THE SCHOOLS OF MEDIAEVAL ENG- 
LAND. SHRINES OF BRITISH SAINTS. 



MESSRS. METHUEN'S PUBLICATIONS 



The Arden Shakespeare 

General Editor, R. H. CASE 
Demy 8vo. 6s. net each volume 

An edition of Shakespeare in Single Plays ; each edited with a full Intro- 
duction, Textual Notes, and a Commentary at the foot of the page. 

Classics of Art 

Edited by DR. J. H. W. LAING 
With numerous Illustrations. Wide Royal 8vo 

THE ART OF THE GREEKS, 2is. net. THE net. RAPHAEL, 155. net. REMBRANDT'S 

ART OF THB ROMANS, i6s. net. CHARDIN, ETCHINGS, 315. 6d. net. REMBRANDT'S 

155. net. DONATELLO, i6s. net. GEORGE PAINTINGS, 425. net. TINTORETTO, i6s.net. 

RoMNEY,i5s. net. GHIRLANDAIO, 155. net. TITIAN, i6s. net. TURNER'S SKETCHES AND 

LAWRENCE, 255. net. MICHELANGELO, 15$. DRAWINGS, 155. net. VELASQUEZ, 155. net. 



The * Complete ' Series 

Fully Illustrated. Demy 8vo 



THE COMPLETE AIRMAN, i6s. net. THE 
COMPLETE AMATEUR BOXER, los. 6d. net. 
THE COMPLETE ASSOCIATION FOOT- 
BALLER, xos. 6d. net. THE COMPLETE 
ATHLETIC TRAINER, los. 6d. net. THE 
COMPLETE BILLIARD PLAYER, IDS. 6d. 
net. THE COMPLETE COOK, IDS. 6d. net. 
THE COMPLETE FOXHUNTER, 165. net. 
THE COMPLETE GOLFER, 125. 6d. net. 
THE COMPLETE HOCKEY-PLAYER, zos. 6d. 
net. THE COMPLETE HORSEMAN, 155. 
net. THE COMPLETE JUJITSUAN. Cr. 8vo. 



53. net. THE COMPLETE LAWN TENNIS 
PLAYER, 125. 6d. net. THE COMPLETE 
MOTORIST, los. 6d. net. THE COMPLETE 
MOUNTAINEER, i6s. net. THE COMPLETE 
OARSMAN, 155. net. THE COMPLETE 

PHOTOGRAPHER,I2. 6d. net. THE COMPLETE 

RUGBY FOOTBALLER, ON THE NEW ZEA- 
LAND SYSTEM, 125. 6d. net. THE COM- 
PLETE SHOT, i6s. net. THE COMPLETE 
SWIMMER, los. 6d. net. THE COMPLETE 
YACHTSMAN, i8s. net. 



The Connoisseur's Library 

With numerous Illustrations. Wide Royal 8vo. i us. 6d. net each volume 



ENGLISH COLOURED BOOKS. ETCHINGS. 
EUROPEAN ENAMELS. FINE BOOKS. 
GLASS. GOLDSMITHS' AND SILVERSMITHS' 
WORK. ILLUMINATED MANUSCRIPTS. 



IVORIES. JEWELLERY. MEZZOTINTS. 
MINIATURES. PORCELAIN. SEALS. 
WOOD . SCULPTURE. 



THE DOCTRINE OF THE INCARNATION, 155. 
net. A HISTORY OF EARLY CHRISTIAN 
DOCTRINE, 165. net. INTRODUCTION TO 
THE HISTORY OF RELIGION, 125. 6d. net. 
AN INTRODUCTION TO THE HISTORY OF 



Handbooks of Theology 

Demy 8vo 

THE CREEDS, 125. 6d. net. THE PHILOSOPHY 
OF RELIGION IN ENGLAND AND AMERICA, 
I2s. 6d. net. THE XXXIX ARTICLES or 
THE CHURCH OF ENGLAND, 15$. net. 



Health Series 

Fcap. 8vo. 25. 6d. net 



THE BABY. THE CARE OF THE BODY. THE 
CARE OF THE TEETH. THE EYES OF OUR 
CHILDREN. HEALTH FOR THE MIDDLE- 
AGED. THE HEALTH OF A WOMAN. THE 
HEALTH OF THE SKIN. How TO LIVE 



LONG. THE PREVENTION OF THE COMMON 
COLD. STAYING THE PLAGUE. THROAT 
AND EAR TROUBLES. TUBERCULOSIS. 
THE HEALTH OF THE CHILD, 2$. net. 



6 MESSRS. METHUEN'S PUBLICATIONS 

The Library of Devotion 

Handy Editions of the great Devotional Books, vrell edited 
With Introductions and (where necessary) Notes 
Small Pott 8vo, cloth, 35. net and 35. >d. net 

Little Books on Art 

With many Illustrations. Demy i6wo. 55. net each volume 
Each volume consists of about 200 pages, and contains from 30 to 40 

Illustrations, including a Frontispiece in Photogravure 
ALBRECKT DCRKR. THE ARTS OF JAPAN. 



BOOKPLATES. BOTTICELLI. BURNE- 
JONES. CELLINI. CHRISTIAN SYMBOLISM. 
CHRIST IN ART. CLAUDE. CONSTABLE. 
COROT. EARLY ENGLISH WATER-COLOUR. 
ENAMELS. FREDERIC LEIGHTON. GEORGE 
ROMNEY. GREEK ART. GREUZE AND 



BOUCHER. HOLBEIN. ILLUMINATED 
MANUSCRIPTS. JEWELLERY. JOHN HOPP- 
NER. Sir JOSHUA REYNOLDS. MILLET. 
MINIATURES. OUR LADY IN ART. 
RAPHAEL. RODIN. TURNER. VANDYCK. 
WATTS. 



The Little Guides 

With many Illustrations by E. H. NEW and other artists, and from 
photographs 

Small Pott Svo. 45. net to 75. 6d. net. 
Guides to the English and Welsh Counties, and some well-known districts 

The main features of these Guides are (i) a handy and charming form ; 
(2) illustrations from photographs and by well-known artists ; (3) good 
plans and maps ; (4) an adequate but compact presentation of everything 
that is interesting in the natural features, history, archaeology, and archi- 
tecture of the town or district treated. 

The Little Quarto Shakespeare 

Edited by W. J. CRAIG. With Introductions and Notes 

Pott i6mo. 40 Volumes. Leather, price is. gd. net each volume 
Cloth, is. 6d. net. 

Plays 

Fcap. Svo. 35. 6d. net 



MILESTONES. Arnold Bennett and Edward 
Knoblock. Tenth Edition. 

IDEAL HUSBAND, AN. Oscar Wilde. Act- 
ing Edition. 

KISMET. Edward Knoblock. Fourth Edi- 
tion. 

THE GREAT ADVENTURE. Arnold Bennett. 
Fifth Edition. 



TYPHOON. A Play in Four Acts. Melchior 

Lengyel. English Version by Laurence 

Irving. Second Edition. 
WARE CASE, THE. George Pleydell. 
GENERAL POST. J. E. Harold Terry. 

Second Edition. 
THE HONEYMOON. Arnold Bennett. Third 

Edition. 



MESSRS. METHUEN'S PUBLICATIONS 



Sports 

Illustrated. 

ALL ABOUT FLYINO, 35. net. ALPINE 
SKI-ING AT ALL HEIGHTS AND SEASONS, 
55. net. CROSS COUNTRY SKI-ING, 55. net. 
GOLF Do's AND DONT'S, 2s. 6d. net. 
QUICK CUTS TO GOOD GOLF, as. 6d. net. 
INSPIRED GOLF, as. 6d. net. DRIVING. 
APPROACHING, PUTTING, as. net. GOLF 
CLUBS AND How TO USE THEM, ss. net. 
THE SECRET OF GOLF FOR OCCASIONAL 



Series 

Fcap. 8vo 

PLAYERS, as. net. LAWN TENNIS, 3*. net. 
LAWN TENNIS Do's AND DONT'S, as. net. 
LAWN TENNIS FOR YOUNG PLAYEIS, 
as. 6d. net. LAWN TENNIS FOR CLUB 
PLAYERS, as. 6d. net. LAWN TENNIS FOR 
MATCH PLAYERS, zs. 6d. net. HOCKEY, 
45. net. How TO SWIM, 2s. net. PUNT- 
ING, 35. 6d. net. SKATING, 35. net. 
WRESTLING, as. net. 



The Westminster Commentaries 

General Editor, WALTER LOCK 

Demy 8vo 

THE ACTS o THB APOSTLES, ias. 6d. net. 
I CORINTHIANS, 8s. 



AMOS, 8s. 6d. net. 

6d. net. EXODUS, 155. net. EZEKIEL, 
12$. 6d. net. GENESIS, i6s. net. HEBREWS, 
8s. 6d. net. ISAIAH, 16$. net. JEREMIAH, 



i6s.net. JoB.8s.6d.net. THE PASTORAL 
EPISTLES, 8s. 6d. net. THE PHILIPPIANS, 
8s. 6d. net. ST. JAMES, 8s. 6d. net. ST. 
MATTHEWS, 155. net. ST. LUKE, 155. net. 



Methuen's Two-Shilling Library 

Cheap Editions of many Popular Books 
Fcap. 8vo 

PART III. A SELECTION OF WORKS OF FICTION 



Bennett (Arnold) 

CLAYHANGER, 8s. net. HILDA LESSWAYS, 
8s. 6d. net. THESE TWAIN. THE CARD. 
THB REGENT : A Five Towns Story of 
Adventure in London. THE PRICE OF 
LOVE. BURIED ALIVE. A MAN FROM 
THE NORTH. THE MATADOR OF THE FIVE 
TOWNS. WHOM GOD HATH JOINED. A 
GREAT MAN : A Frolic. MR. PROHACK. 
All 7$. 6d. net. 

Birmingham (George A.) 

SPANISH GOLD. THE SEARCH PARTY. 
LALAGK'S LOVERS. THE BAD TIMES. UP, 
THE REBELS. THE LOST LAWYER. All 
7s. 6d. net. INISKE.ENY, 8s. 6d. net. 

Burroughs (Edgar Rice) 
TARZAN OF THE APES, 6s. net. THE 
RETURN OF TARZAN, 6s. net. THE BEASTS 
OF TARZAN, 6s. net. THE SON OF TARZAN, 
6s. net. JUNGLE TALES OF TARZAN, 6s. 
net. TARZAN AND THE JEWELS OF OPAR, 
6s. net. TARZAN THE UNTAMED, 7s. 6d. net. 
A PRINCESS OF MARS, 6s. net. THE GODS 
OF MARS, 6s. net. THE WARLORD OF 
MARS, 6s. net. THUVIA, MAID OF MARS, 
6s. net. TARZAN THE TERRIBLE, as. 6d. net. 
THE MUCKER, 6s. net. THE MAN WITH- 
OUT A SOUL, 6s. net. 

Conrad (Joseph) 

A SET OF Six, 7s. 6d. net. VICTORY : An 
Island Tale. Cr. 8vo. gs. net. THE 
SECRET AGENT : A Simple Tale. Cr. 8vo. 
gs. net. UNDER WESTERN EYES. Cr. 
Hvo. gs. net. CHANCE. Cr. 8vo. gs. net. 



Corelli (Marie) 

A ROMANCE OF Two WORLDS, 7s. 6d. net. 
VENDETTA : or, The Story of One For- 
gotten, 8s. net. THELMA : A Norwegian 
Princess, 8s. 6d. net. ARDATH : The Story 
of a Dead Self, 7s. 6d. net. THE SOUL OF 
LILITH, 75. 6d. net. WORMWOOD : A Drama 
of Paris, 8s. net. BARABBAS : A Dream of 
the World's Tragedy, 8s. net. THE SORROWS 
OF SATAN, 7s. 6d. net. THE MASTER- 
CHRISTIAN, 8s. 6d. net. TEMPORAL POWER: 
A Study in Supremacy, 6s. net. GOD'S 
GOOD MAN : A Simple Love Story, 8s. 6d. 
net. HOLY ORDERS : The Tragedy of a 
Quiet Life, 8s. 6d. net. THE MIGHTY ATOM, 
7s. 6d. net. BOY : A Sketch, 7s. 6d. net. 
CAMEOS, 6s. net. THE LIFE EVERLASTING, 
8s. 6d. net. THE LOVB OF LONG AGO, AND 
OTHER STORIES, 8s. 6d. net. INNOCENT, 
7s. 6d. net. THE SECRET POWER : A 
Romance of the Time, 7$. 6d. net. 



Hichens (Robert) 

TONGUES OF CONSCIENCB, 7s. 6d. ntt. 
FELIX : Three Years in a Life, 7s. 6d. net. 
THE WOMAN WITH THE FAN, 7s. 6d. net. 
THE GARDEN OF ALLAH, 8s. 6d. net. 
THE CALL OF THE BLOOD, 8s. 6d. net. 
THE DWELLER ON THE THRESHOLD, 7s. 6d. 
net. THE WAY OF AMBITION, 7s. 6d. net. 
IN THE WILDERNESS, 7s. 6d. v*t. 



MESSRS. METHUEN'S PUBLICATIONS 



Hope (Anthony) 

A CHANGE OF AIR. A MAN OF MARK. 
SIMON DALE. THE KING'S MIRROR. 
THE DOLLY DIALOGUES. MRS. MAXON 
PROTESTS. A YOUNG MAN'S YEAR. 
BEAUMAROY HOME FROM THE WARS. 
All 7s. 6d. net. 

Jacobs (W. W.) 

MANY CARGOES, 5*. net. SEA URCHINS, 
5$. net and 35. 6d. net. A MASTER OF 
CRAFT, 6s. net. LIGHT FREIGHTS, 55. net. 
THE SKIPPER'S WOOING, 55. net. AT SUN- 
WICH PORT, 55. net. DIALSTONE LANE, 
55. net. ODD CRAFT, 55. net. THE LADY 
OF THE BARGE, 55. net. SALTHAVEN, 55. 
net. SAILORS' KNOTS, 55. net. SHORT 
CRUISES, 6s. net. 

London (Jack) WHITE FANG. Ninth 
Edition. Cr. 8vo. 7s. 6d. net. 

Lucas (E. V.) 

LISTENER'S LURE : An Oblique Narration, 
6s. net. OVER BEMERTON'S : An Easy- 
going Chronicle, 6s. net. MR. INGLESIDE, 
6s. net. LONDON LAVENDER, 6s. net. 
LANDMARKS, 6s. net. THE VERMILION 
Box, 6s. net. VERENA IN THE MIDST, 
8s. 6d. net. ROSE AND PO S E, 6s. net. 

McKenna (Stephen) 
SONIA : Between Two Worlds, 8s. net. 
NINETY-SIX HOURS' LEAVE, 7s. 6d. net. 
THE SIXTH SENSE, 6s. net. MIDAS & SON, 
8s. net. 

Malet (Lucas) 

THE HISTORY OF SIR RICHARD CALMADY : 
A Romance. IDS. net. THE CARISSIMA. 
THE GATELESS BARRIER. DEADHAM 
HARD. All 7s. 6d. net. THE WAGES OF 
SIN. 8s. net. COLONEL ENDERBY'S WIFE, 
73. 6d. net. 

Mason (A. E. W.). CLEMENTINA. 
Illustrated. Ninth Edition. Cr. 8vo. 
7s. 6d. net. 

Milne (A. A.) 

THE DAY'S PLAY. THE HOLIDAY ROUND. 
ONCE A WEEK. AH Cr. 8vo. 7$. 6d. net. 
THE SUNNY SIDE. Cr. 8vo. 6s. net. 
THE RED HOUSE MYSTERY. Cr. 8vo. 
6s. net. 



Oxenham (John) 

PROFIT AND Loss. THK SONG OF HYA- 
CINTH, and Other Stories. THE COIL OF 
CARNE. THE QUEST OF THE GOLDEN Ross. 
MARY ALL-ALONE. All7s.6d.net. 

Parker (Gilbert) 

MRS. FALCHION. THE TRANSLATION 
OF A SAVAGE. WHEN VALMOND CAME 
TO PONTIAC : The Story bf a Lost 
Napoleon. AN ADVENTURE OF THE 
NORTH : The Last Adventures of ' Pretty 
Pierre.' THE SEATS OF THE MIGHTY. THE 
BATTLE OF THE STRONG : A Romance 
of Two Kingdoms. THE TRAIL OF THE 
SWORD. NORTHERN LIGHTS. All 75. 6d. 
net. 

Phillpotts (Eden) 

CHILDREN OF THE MIST. THE RIVER. 
DEMETER'S DAUGHTER. THE HUMAN 
BOY AND THE WAR. AH 7s. 6d. net. 

Rohmer (Sax) 

TALES OF SECRET EGYPT. THE ORCHARD 
OF TEARS. THE GOLDEN SCORPION. All 
7s. 6J. net. THE DEVIL DOCTOR. 
THE MYSTERY OF DR. FU-MANCHU. THE 
YELLOW CLAW. All 35. 6d. net. 

Swinnerton (F.) SHOPS AND HOUSES. 
SEPTEMBER. THE HAPPY FAMILY. ON 
THE STAIRCASE. COQUETTE. THE CHASTE 
WIFE. All 7s. 6d. net. THE MERRY 
HEART, THE CASEMENT, THE YOUNG 
IDEA. All 6s. net. 

Wells (H. G.). BEALBY. Fourth Edition. 
Cr. 8vo. 75 6d. net. 

Williamson (C. N. and A. M.)-- 

THE LIGHTNING CONDUCTOR : The Strange 
Adventures of a Motor Car. LADY BETTY 

ACROSS THE WATER. I T HAPPENED IN 

EGYPT. THE SHOP GIRL. THE LIGHTNING 
CONDUCTRESS. MY FRIEND THE 

CHAUFFEUR. SET IN SILVER. THE 
GREAT PEARL SECRET. THE LOVE 
PIRATE. All 7s. 6d. net. CRUCIFIX 
CORNER. 6s. net. 



7 22 



Methuen's Two-Shilling Novels 

Cheap Editions of many of the most Popular Novels of the day 

Write for Complete List 

Fcap. 8vo 



THIS BOOK IS DUE ON THE LAST DATE 
STAMPED BELOW 



T! 



AN INITIAL FINE OF 25 CENTS 

WILL BE ASSESSED FOR FAILURE TO RETURN 
THIS BOOK ON THE DATE DUE. THE PENALTY 
WILL INCREASE TO SO CENTS ON THE FOURTH 
DAY AND TO $1.OO ON THE SEVENTH DAY 
OVERDUE. 



. 




47 






j-? 


[op 


17/ 


nrr. * 7 w* 6 








DEC 5 1949$ 


w 


C , ~k \ c &&-~ 




\ 
























- 






















10m-7,'44(1064s) 




<^- 




Engineering 
Library 



UNIVERSITY OF CALIFORNIA LIBRARY